diff --git a/application/AMS Cover.pdf b/application/AMS Cover.pdf new file mode 100644 index 00000000..2773861d Binary files /dev/null and b/application/AMS Cover.pdf differ diff --git a/application/cover letter teaching/Anton_Bobkov Agnes Scott College.pdf b/application/cover letter teaching/Anton_Bobkov Agnes Scott College.pdf new file mode 100644 index 00000000..5e7cf023 Binary files /dev/null and b/application/cover letter teaching/Anton_Bobkov Agnes Scott College.pdf differ diff --git a/application/cover letter teaching/Anton_Bobkov CUNY.pdf b/application/cover letter teaching/Anton_Bobkov CUNY.pdf new file mode 100644 index 00000000..48bb0d75 Binary files /dev/null and b/application/cover letter teaching/Anton_Bobkov CUNY.pdf differ diff --git a/application/cover letter teaching/Anton_Bobkov Kalamazoo update.pdf b/application/cover letter teaching/Anton_Bobkov Kalamazoo update.pdf new file mode 100644 index 00000000..fd59066c Binary files /dev/null and b/application/cover letter teaching/Anton_Bobkov Kalamazoo update.pdf differ diff --git a/application/cover letter teaching/Anton_Bobkov Pomona College.pdf b/application/cover letter teaching/Anton_Bobkov Pomona College.pdf new file mode 100644 index 00000000..4312846f Binary files /dev/null and b/application/cover letter teaching/Anton_Bobkov Pomona College.pdf differ diff --git a/application/cover letter teaching/Anton_Bobkov USC.pdf b/application/cover letter teaching/Anton_Bobkov USC.pdf new file mode 100644 index 00000000..978b9646 Binary files /dev/null and b/application/cover letter teaching/Anton_Bobkov USC.pdf differ diff --git a/application/cover letter teaching/Anton_Bobkov.tex b/application/cover letter teaching/Anton_Bobkov.tex index 6bc8bd2f..8ce695ef 100644 --- a/application/cover letter teaching/Anton_Bobkov.tex +++ b/application/cover letter teaching/Anton_Bobkov.tex @@ -26,6 +26,7 @@ Unfortunately I will not be able to attend the joint AMS/MAA meetings this year.}} \def\recommendation{{my advisor, Matthias Aschenbrenner, as well as from Artem Chernikov, Olga Radko, and Jukka Virtanen. Dr.~Radko's and Dr.~Virtanen's letters address my teaching experience}} +\def\three{{my advisor, Matthias Aschenbrenner, as well as from Olga Radko and Jukka Virtanen. Dr.~Radko's and Dr.~Virtanen's letters address my teaching experience}} \def\teachrecommendation{{Olga Radko and Jukka Virtanen addressing my teaching experience}} \def\mathjobs{{, as posted on the MathJobs website.}} @@ -37,8 +38,83 @@ \def\genintro{{To the Academic Hiring Committee:}} \def\lecture{{a lecturer position in mathematics}} + +\coverletter + {} + {\textbf{Update:} I am interested in a full time non tenure track visiting appointment for a term of one year (with possible extension to two years).\\ \ \\ \genintro} + {an assistant professor position (for both tenure-track and non-tenure-track appointments)} + {\mathjobs} + {\teachliberalart} + {the Kalamazoo College} + {\generic, as well as undergraduate and graduate transcripts} + {\recommendation} +\end{document} + + +\coverletter + {} + {\genintro} + % {a Visiting Assistant Professor position} + % {a Mathematics Instructor position} + {a Lecturer position} + {\mathjobs} + {} + % {\teachliberalart} + {the City College of New York} + {\noresearch, as well as a diversity statement and a teaching portfolio} + % {\noresearch, as well as a graduate transcript} + % {\publication, as well as a graduate transcript} + % {\publication, as well as a teaching portfolio} + % {\noresearch} + % {\generic, as well as a graduate transcript and a diversity statement} + % {\generic} + % {\recommendation} + % {\teachrecommendation} + {\three} +\end{document} + +\coverletter + {} + {\genintro} + % {a Visiting Assistant Professor position} + {Lecture, Assistant Professor, Associate Professor, and Professor of Mathematics positions} + % {a Mathematics Instructor position} + {\mathjobs} + {} + % {\teachliberalart} + {the University of Southern California} + % {\noresearch, as well as a graduate transcript} + % {\publication, as well as a graduate transcript} + {\publication, as well as a teaching portfolio} + % {\noresearch} + % {\generic, as well as a graduate transcript and a diversity statement} + % {\generic} + {\recommendation} + % {\teachrecommendation} +\end{document} + + +\coverletter + {} + {\genintro} + {a Visiting Assistant Professor position} + % {a Mathematics Instructor position} + {\mathjobs} + % {} + {\teachliberalart} + {the Pomona College} + % {\noresearch, as well as a graduate transcript} + {\publication, as well as a graduate transcript} + % {\noresearch} + % {\generic, as well as a graduate transcript and a diversity statement} + % {\generic} + {\recommendation} + % {\teachrecommendation} +\end{document} + + \coverletter {} {\genintro} @@ -223,17 +299,6 @@ {\recommendation} \end{document} -\coverletter - {} - {\genintro} - {an assistant professor position (for both tenure-track and non-tenure-track appointments)} - {\mathjobs} - {\teachliberalart} - {the Kalamazoo College} - {\generic, as well as undergraduate and graduate transcripts} - {\recommendation} -\end{document} - \coverletter {} diff --git a/application/merged_document.pdf b/application/merged_document.pdf new file mode 100644 index 00000000..32b87cfa Binary files /dev/null and b/application/merged_document.pdf differ diff --git a/application/teaching portfolio/diversity_statement.pdf b/application/teaching portfolio/diversity_statement.pdf index b30483f1..03dd109e 100644 Binary files a/application/teaching portfolio/diversity_statement.pdf and b/application/teaching portfolio/diversity_statement.pdf differ diff --git a/application/teaching portfolio/diversity_statement.tex b/application/teaching portfolio/diversity_statement.tex index 421c762c..be88a54c 100644 --- a/application/teaching portfolio/diversity_statement.tex +++ b/application/teaching portfolio/diversity_statement.tex @@ -7,7 +7,7 @@ %\usepackage{mathrsfs} \usepackage{setspace} -% \setstretch{3} +% \setstretch{3} % \doublespacing \usepackage [english]{babel} diff --git a/application/teaching portfolio/teaching_portfolio.pdf b/application/teaching portfolio/teaching_portfolio.pdf index 81d3e82f..4115de40 100644 Binary files a/application/teaching portfolio/teaching_portfolio.pdf and b/application/teaching portfolio/teaching_portfolio.pdf differ diff --git a/application/teaching portfolio/teaching_portfolio.tex b/application/teaching portfolio/teaching_portfolio.tex index 029108d2..cf62f933 100644 --- a/application/teaching portfolio/teaching_portfolio.tex +++ b/application/teaching portfolio/teaching_portfolio.tex @@ -5,7 +5,7 @@ % % This template has been downloaded from: % http://www.LaTeXTemplates.com -% +% % Original author: % Rensselaer Polytechnic Institute (http://www.rpi.edu/dept/arc/training/latex/resumes/) % diff --git a/research/02 Trees vc-density/Trees_vc_density.pdf b/research/02 Trees vc-density/Trees_vc_density.pdf index 28967cfe..03727326 100644 Binary files a/research/02 Trees vc-density/Trees_vc_density.pdf and b/research/02 Trees vc-density/Trees_vc_density.pdf differ diff --git a/research/02 Trees vc-density/Trees_vc_density.tex b/research/02 Trees vc-density/Trees_vc_density.tex index 052d58ac..ee0c3cd5 100644 --- a/research/02 Trees vc-density/Trees_vc_density.tex +++ b/research/02 Trees vc-density/Trees_vc_density.tex @@ -2,6 +2,7 @@ \usepackage{../AMC_style} \usepackage{../Research} +\usepackage{../Thm} \usepackage{tikz} @@ -65,18 +66,18 @@ \title{vc-density for trees} \author{Anton Bobkov} \email{bobkov@math.ucla.edu} -%more info +% more info \begin{abstract} - We show that for the theory of infinite trees we have $\vc(n) = n$ for all $n$. + We show that for the theory of infinite trees we have $\vc(n) = n$ for all $n$. \end{abstract} \maketitle -VC-density was studied in \cite{vc_density} by Aschenbrenner, Dolich, Haskell, MacPherson, and Starchenko as a natural notion of dimension for NIP theories. In an NIP theory we can define a vc-function +VC-density was studied in \cite{density} by Aschenbrenner, Dolich, Haskell, MacPherson, and Starchenko as a natural notion of dimension for NIP theories. In an NIP theory we can define a vc-function \begin{align*} - \vc : \N \arr \N + \vc : \N \arr \N \end{align*} Where $\vc(n)$ measures the worst-case complexity of definable sets in an $n$-dimensional space. Simplest possible behavior is $\vc(n) = n$ for all $n$. Theories with the property that $\vc(1) = 1$ are known to be dp-minimal, i.e. having the smallest possible dp-rank. In general, it is not known whether there can be a dp-minimal theory which doesn't satisfy $\vc(n)=n$. @@ -85,10 +86,10 @@ % More precisely, our structure is branches of an infinite tree in a langugage of $\leq$ with possibly finitely many colors. Parigot in \cite{parigot_trees} showed that such structures have NIP. This result was strengthened by Simon in \cite{simon_dp_min} showing that trees are dp-minimal. -The paper \cite{vc_density} poses the following problem: +The paper \cite{density} poses the following problem: -\begin{Problem} (\cite{vc_density} p. 47) - Determine the VC density function of each (infinite) tree. +\begin{Problem} (\cite{density} p. 47) + Determine the VC density function of each (infinite) tree. \end{Problem} Here we settle this question by showing any model of trees has $\vc(n) = n$. @@ -104,9 +105,9 @@ The language for the trees consists of a single binary predicate $\{\leq\}$. The theory of trees states that $\leq$ defines a partial order and for every element $a$ we have that $\{x \mid x < a\}$ is a linear order. For visualization purposes we assume that trees grow upwards, with the smaller elements on the bottom and the larger elements on the top. If $a \leq b$ we will say that $a$ is below $b$ and $b$ is above $a$. \begin{Definition} - Work in a tree $\TT = (T, \leq)$. - For $x \in T$ let $I(x) = \{t \in T \mid t \leq x\}$ denote all the elements below $x$. - The \emph{meet} of two tree elements $a,b$ is the greatest element of $I(a) \cap I(b)$ (if one exists) and is denoted by $a \wedge b$. + Work in a tree $\TT = (T, \leq)$. + For $x \in T$ let $I(x) = \{t \in T \mid t \leq x\}$ denote all the elements below $x$. + The \emph{meet} of two tree elements $a,b$ is the greatest element of $I(a) \cap I(b)$ (if one exists) and is denoted by $a \wedge b$. \end{Definition} The theory of meet trees requires that any two elements in the same connected component have a meet. Colored trees are trees with a finite number of colors added via unary predicates. @@ -114,169 +115,22 @@ From now on assume that all trees are colored. We allow our trees to be disconnected (so really, we work with forests) or finite unless otherwise stated. -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{VC-dimension and vc-density} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{Definition} - Throughout this section we work with a collection $\F$ of subsets of a set $X$. - We call the pair $(X, \F)$ a \defn{set system}. - \begin{itemize} - \item Given a subset $A$ of $X$, we define the set system $(A, A \cap \F)$ - where $A \cap \F = \curly{A \cap F}_{F\in \F}$. - \item For $A \subset X$ we say that $\F$ \defn{shatters} $A$ if $A \cap \F = \PP(A)$. - \end{itemize} -\end{Definition} - -\begin{Definition} - We say $(X, \F)$ has VC-dimension $n$ if the largest subset of $X$ shattered by $\F$ is of size $n$. - If $\F$ shatters arbitrarily large subsets of $X$, we say that $(X, \F)$ has infinite VC-dimension. - We denote the VC-dimension of $(X, \F)$ by $\VC(\F)$. -\end{Definition} - -\begin{Note} - We may drop $X$ from the previous definition, as the VC-dimension doesn't depend on the base set and is determined by $(\bigcup \F, \F)$. -\end{Note} -This allows us to distinguish between well-behaved set systems of finite VC-dimension which tend to have good combinatorial properties and -poorly behaved set systems with infinite VC-dimension. - -Another natural combinatorial notion is that of a dual system: -\begin{Definition} - For $a \in X$ define $X_a = \curly{F \in \F \mid a \in F}$. - Let $\F^* = \curly{X_a}_{a \in X}$. - We define $(\F, \F^*)$ as the \defn{dual system} of $(X, \F)$. - The VC-dimension of the dual system of $(X, \F)$ is referred to as the \defn{dual VC-dimension} of $(X, \F)$ and denoted by $\VC^*(\F)$. - (As before, this notion doesn't depend on $X$.) -\end{Definition} - -\begin{Lemma} - A set system has finite VC-dimension if and only if its dual system has finite VC-dimension. - More precisely - \begin{align*} - \VC^*(\F) \leq 2^{1+\VC(\F)}. - \end{align*} -\end{Lemma} - -For a more refined notion we look at the traces of our family on finite sets: -\begin{Definition} - Define the \defn{shatter function} $\pi_\F \colon \N \arr \N$ and the \defn{dual shatter function} $\pi^*_\F \colon \N \arr \N$ of $\F$ by - \begin{align*} - \pi_\F(n) &= \max \curly{|A \cap \F| \mid A \subset X \text{ and } |A| = n} \\ - \pi^*_\F(n) &= \max \curly{\text{atoms($B$)} \mid B \subset \F, |B| = n} - \end{align*} - where atoms($B$) = number of atoms in the Boolean algebra generated by $B$. - Note that the dual shatter function is precisely the shatter function of the dual system: $\pi^*_\F = \pi_{\F^*}$. -\end{Definition} - -A simple upper bound is $\pi_\F(n) \leq 2^n$ (same for the dual). -If VC-dimension is infinite then clearly $\pi_\F(n) = 2^n$ for all $n$. Conversely we have the following remarkable fact: -\begin{Theorem} [Sauer-Shelah '72] - If the set system $(X, \F)$ has finite VC-dimension $d$ then $\pi_\F(n) \leq {n \choose \leq d}$ where - ${n \choose \leq d} = {n \choose d} + {n \choose d - 1} + \ldots + {n \choose 1}$. -\end{Theorem} - -Thus the systems with a finite VC-dimension are precisely the systems where the shatter function grows polynomially. -Define vc-density to be the degree of that polynomial: -\begin{Definition} - Define \defn{vc-density} and \defn{dual vc-density} of $\F$ as - \begin{align*} - \vc(\F) &= \limsup_{n \to \infty}\frac{\log \pi_\F(n)}{\log n} \in \R^{\geq 0} \cup \curly{+\infty}\\ - \vc^*(\F) &= \limsup_{n \to \infty}\frac{\log \pi^*_\F(n)}{\log n}\in \R^{\geq 0} \cup \curly{+\infty} - \end{align*} -\end{Definition} - -Generally speaking a shatter function that is bounded by a polynomial doesn't itself have to be a polynomial. -Proposition 4.12 in \cite{vc_density} gives an example of a shatter function that grows like $n \log n$ (so it has vc-density $1$). - -So far the notions that we have defined are purely combinatorial. -We now adapt VC-dimension and vc-density to the model theoretic context. - -\begin{Definition} - Work in a structure $M$. - Fix a finite collection of formulas $\Phi(x, y) = \curly{\phi_i(x, y)}$. - - \begin{itemize} - \item For $\phi(x, y) \in \LL(M)$ and $b \in M^{|y|}$ let - \begin{align*} - \phi(M^{|x|}, b) = \{a \in M^{|x|} \mid \phi(a, b)\} \subseteq M^{|x|}. - \end{align*} - \item Let $\Phi(M^{|x|}, M^{|y|})= \{\phi_i(M^{|x|}, b) \mid \phi_i \in \Phi, b \in M^{|y|}\} \subseteq \PP(M^{|x|})$. - \item Let $\F_\Phi = \Phi(M^{|x|}, M^{|y|})$ giving a set system $(M^{|x|}, \F_\Phi)$. - \item Define \defn{VC-dimension} of $\Phi$, $\VC(\Phi)$ to be the VC-dimension of $(M^{|x|}, \F_\Phi)$, similarly for the dual. - \item Define \defn{vc-density} of $\Phi$, $\vc(\Phi)$ to be the vc-density of $(M^{|x|}, \F_\Phi)$, similarly for the dual. - \end{itemize} - - We will also refer to the vc-density and VC-dimension of a single formula $\phi$ - viewing it as a one element collection $\curly{\phi}$. -\end{Definition} - -Counting atoms of a Boolean algebra in a model theoretic setting corresponds to counting types, -so it is instructive to rewrite the shatter function in terms of types. - -\begin{Definition} - \begin{align*} - \pi^*_\Phi(n) &= \max \curly{\text{number of $\Phi$-types over $B$} \mid B \subset M, |B| = n} - \end{align*} -\end{Definition} - -\begin{Lemma} \label{count_types} - \begin{align*} - \vc^*(\Phi) &= \text{degree of polynomial growth of $\pi^*_\Phi(n)$} = \limsup_{n \to \infty}\frac{\log \pi^*_\Phi(n)}{\log n} - \end{align*} -\end{Lemma} - -One can check that the shatter function and hence VC-dimension and vc-density of a formula are elementary notions, -so they only depend on the first-order theory of the structure. - -NIP theories are a natural context for studying vc-density. -In fact we can take the following as the definition of NIP: -\begin{Definition} - Define $\phi$ to be NIP if it has finite VC-dimension. -\end{Definition} - -% \cite{Aschenbrenner_reference_8} shows that in a general combinatorial context, -In a general combinatorial context, -vc-density can be any real number in $0 \cup [1, \infty)$. -Less is known if we restrict our attention to NIP theories. -Proposition 4.6 in \cite{vc_density} gives examples of formulas that have non-integer rational vc-density in an NIP theory, -however it is open whether one can get an irrational vc-density in this context. - -In general, instead of working with a theory formula by formula, we can look for a uniform bound for all formulas: -\begin{Definition} - For a given NIP structure $M$, define the \defn{vc-function} - \begin{align*} - \vc^M(n) &= \sup \{\vc^*(\phi(x, y)) \mid \phi \in \LL(M), |x| = n\} \\ - &= \sup \{\vc(\phi(x, y)) \mid \phi \in \LL(M), |y| = n\} - \end{align*} -\end{Definition} - -As before this definition is elementary, so it only depends on the theory of $M$. -We omit the superscript $M$ if it is understood from the context. -One can easily check the following bounds: -\begin{Lemma} [Lemma 3.22 in \cite{vc_density}] - \begin{align*} - \vc(1) &\geq 1 \\ - \vc(n) &\geq n\vc(1) - \end{align*} -\end{Lemma} - -However, it is not known whether the second inequality can be strict or even whether $\vc(1) < \infty$ implies $\vc(n) < \infty$. +\input{../vc_intro.tex} \section{Proper Subdivisions: Definition and Properties} We work with finite relational languages. Given a formula we define its complexity as the depth of quantifiers used to build up the formula. More precisely: -%See for example \cite{ynm_notes} Definition 2D.4 pg.72. +% See for example \cite{ynm_notes} Definition 2D.4 pg.72. \begin{Definition} -Define \emph{complexity} of a formula by induction: -\begin{align*} - &\cx(\text{q.f. formula}) = 0 \\ - &\cx(\exists x \phi(x)) = \cx(\phi(x)) + 1 \\ - &\cx(\phi \wedge \psi) = \max(\cx(\phi), \cx(\psi)) \\ - &\cx(\neg \phi) = \cx(\phi) -\end{align*} + Define \emph{complexity} of a formula by induction: + \begin{align*} + &\cx(\text{q.f. formula}) = 0 \\ + &\cx(\exists x \phi(x)) = \cx(\phi(x)) + 1 \\ + &\cx(\phi \wedge \psi) = \max(\cx(\phi), \cx(\psi)) \\ + &\cx(\neg \phi) = \cx(\phi) + \end{align*} \end{Definition} A simple inductive argument verifies that there are (up to equivalence) only finitely many formulas when the complexity and the number of free variables are fixed. We will use the following notation for types: @@ -312,25 +166,25 @@ \section{Proper Subdivisions: Definition and Properties} \end{Definition} \begin{Lemma} \label{lm_subdivision} - Consider a subdivision $(\A, \B)$ of $\TT$. If $(\A, \B)$ is $0$-proper then it is proper. + Consider a subdivision $(\A, \B)$ of $\TT$. If $(\A, \B)$ is $0$-proper then it is proper. \end{Lemma} \begin{proof} - We prove that the subdivision is $k$-proper for all $k$ by induction. - The case $k = 0$ is given by the assumption. - Suppose we have $\TT \models \exists x \, \phi^k(x, a_1, b_1)$ where $\phi^k$ is some formula of complexity $k$. Let $a \in T$ witness the existential claim, i.e., $\TT \models \phi^k(a, a_1, b_1)$. We can have $a \in A$ or $a \in B$. Without loss of generality assume $a \in A$. Let $\pp = \tp^k_{\A} (a, a_1)$. Then we have - \begin{align*} - \A \models \exists x \, \tp^k_{\A}(x, a_1) = \pp - \end{align*} - (with $\tp^k_{\A}(x, a_1) = \pp$ a shorthand for $\phi_{\pp}(x)$ where $\phi_{\pp}$ is a formula that determines the type $\pp$). - The formula $\tp^k_{\A}(x, a_1) = \pp$ is of complexity $\leq k$ so $\exists x \, \tp^k_{\A}(x, a_1) = \pp$ is of complexity $\leq k+1$. By the inductive hypothesis we have - \begin{align*} - \A \models \exists x \, \tp^k_{\A}(x, a_2) = \pp. - \end{align*} - Let $a'$ witness this existential claim, so that $\tp^k_{\A}(a', a_2) = \pp$, hence $\tp^k_{\A}(a', a_2) = \tp^k_{\A}(a, a_1)$, that is, - $\A \models a'a_2 \equiv_k aa_1$. By the inductive hypothesis we therefore have - $\TT \models aa_1b_1 \equiv_k a'a_2b_2$; in particular $\TT \models \phi^k(a', a_2, b_2) \text {as } \TT \models \phi^k(a, a_1, b_1)$, - and $\TT \models \exists x \phi^k(x, a_2, b_2)$. + We prove that the subdivision is $n$-proper for all $k$ by induction. + The case $n = 0$ is given by the assumption. + Suppose we have $\TT \models \exists x \, \phi^n(x, a_1, b_1)$ where $\phi^n$ is some formula of complexity $n$. Let $a \in T$ witness the existential claim, i.e., $\TT \models \phi^n(a, a_1, b_1)$. We can have $a \in A$ or $a \in B$. Without loss of generality assume $a \in A$. Let $\pp = \tp^n_{\A} (a, a_1)$. Then we have + \begin{align*} + \A \models \exists x \, \tp^n_{\A}(x, a_1) = \pp + \end{align*} + (with $\tp^n_{\A}(x, a_1) = \pp$ a shorthand for $\phi_{\pp}(x)$ where $\phi_{\pp}$ is a formula that determines the type $\pp$). + The formula $\tp^n_{\A}(x, a_1) = \pp$ is of complexity $\leq k$ so $\exists x \, \tp^n_{\A}(x, a_1) = \pp$ is of complexity $\leq k+1$. By the inductive hypothesis we have + \begin{align*} + \A \models \exists x \, \tp^n_{\A}(x, a_2) = \pp. + \end{align*} + Let $a'$ witness this existential claim, so that $\tp^n_{\A}(a', a_2) = \pp$, hence $\tp^n_{\A}(a', a_2) = \tp^n_{\A}(a, a_1)$, that is, + $\A \models a'a_2 \equiv_n aa_1$. By the inductive hypothesis we therefore have + $\TT \models aa_1b_1 \equiv_n a'a_2b_2$; in particular $\TT \models \phi^n(a', a_2, b_2) \text {as } \TT \models \phi^n(a, a_1, b_1)$, + and $\TT \models \exists x \phi^n(x, a_2, b_2)$. \end{proof} This lemma is general, but we will use it specifically applied to (colored) trees. @@ -348,26 +202,26 @@ \section{Proper Subdivisions: Definition and Properties} \end{Example} \begin{Example} \label{ex_cone} - Fix a tree $\TT$ in the language $\{\leq\}$ and $a \in T$. Let $B = \{t \in T \mid a < t\}$, $S = \{t \in T \mid t \leq a\}$, $A = T - B$. Then $(A, \leq, S)$ and $(B, \leq)$ form a proper subdivision, where $\LL_A$ has a unary predicate interpreted by $S$. -To see this, again, we show that the subdivision is 0-proper. -The only time $a \in A$ and $b \in B$ are comparable is when $a \in S$, and this is captured by the language. -(See proof of Lemma \ref{subdivide} for more details.) + Fix a tree $\TT$ in the language $\{\leq\}$ and $a \in T$. Let $B = \{t \in T \mid a < t\}$, $S = \{t \in T \mid t \leq a\}$, $A = T - B$. Then $(A, \leq, S)$ and $(B, \leq)$ form a proper subdivision, where $\LL_A$ has a unary predicate interpreted by $S$. + To see this, again, we show that the subdivision is 0-proper. + The only time $a \in A$ and $b \in B$ are comparable is when $a \in S$, and this is captured by the language. + (See proof of Lemma \ref{subdivide} for more details.) \end{Example} \begin{Definition} For $\phi(x, y)$, $A \subseteq T^{|x|}$ and $B \subseteq T^{|y|}$ -\begin{itemize} - \item let $\phi(A, b) = \{a \in A \mid \phi(a, b)\} \subseteq A$, and - \item let $\phi(A, B) = \{\phi(A, b) \mid b \in B\} \subseteq \PP(A)$. -\end{itemize} + \begin{itemize} + \item let $\phi(A, b) = \{a \in A \mid \phi(a, b)\} \subseteq A$, and + \item let $\phi(A, B) = \{\phi(A, b) \mid b \in B\} \subseteq \PP(A)$. + \end{itemize} \end{Definition} Thus $\phi(A, B)$ is a collection of subsets of $A$ definable by $\phi$ with parameters from $B$. We notice the following bound when $A, B$ are parts of a proper subdivision. \begin{Corollary} \label{cor_type_count} - Let $\A, \B$ be a proper subdivision of $\TT$ and $\phi(x,y)$ be a formula of complexity $n$. Then $|\phi(A^{|x|}, B^{|y|})|$ is bounded by $|S^n_{\B, |y|}|$. + Let $\A, \B$ be a proper subdivision of $\TT$ and $\phi(x,y)$ be a formula of complexity $n$. Then $|\phi(A^{|x|}, B^{|y|})|$ is bounded by $|S^n_{\B, |y|}|$. \end{Corollary} \begin{proof} - Take some $a \in A^{|x|}$ and $b_1, b_2 \in B^{|y|}$ with $\tp^n_{\B}(b_1) = \tp^n_{\B}(b_2)$. We have $\B \models b_1 \equiv_n b_2$ and (trivially) $\A \models a \equiv_n a$. Thus we have $\TT \models ab_1 \equiv_n ab_2$, so $T \models \phi(a, b_1) \leftrightarrow \phi(a, b_2)$. Since $a$ was arbitrary we have $\phi(A^{|x|}, b_1) = \phi(A^{|x|}, b_2)$ as different traces can only come from parameters of different types. Thus $|\phi(A^{|x|}, B^{|y|}) \leq |S^n_{\B, |y|}|$. + Take some $a \in A^{|x|}$ and $b_1, b_2 \in B^{|y|}$ with $\tp^n_{\B}(b_1) = \tp^n_{\B}(b_2)$. We have $\B \models b_1 \equiv_n b_2$ and (trivially) $\A \models a \equiv_n a$. Thus we have $\TT \models ab_1 \equiv_n ab_2$, so $T \models \phi(a, b_1) \leftrightarrow \phi(a, b_2)$. Since $a$ was arbitrary we have $\phi(A^{|x|}, b_1) = \phi(A^{|x|}, b_2)$ as different traces can only come from parameters of different types. Thus $|\phi(A^{|x|}, B^{|y|}) \leq |S^n_{\B, |y|}|$. \end{proof} We note that the size of the type space $|S^n_{\B, |y|}|$ can be bounded uniformly: @@ -396,256 +250,291 @@ \section{Proper Subdivisions: Constructions} the \emph{closed cone} above $a$. Connected components of that cone can be thought of as \emph{open cones} above $a$. With that interpretation in mind, the notation $E_a(b, c)$ means that $b$ and $c$ are in the same open cone above $a$. More formally: - \begin{align*} - E_a(b, c) \ifff E(b,c) \text{ and } (b \wedge c) > a. - \end{align*} + \begin{align*} + E_a(b, c) \ifff E(b,c) \text{ and } (b \wedge c) > a. + \end{align*} \end{Definition} Fix a language $\LL$ for a colored tree $\LL = \{\leq, C_1, \ldots C_n\} = \{\leq, \vec C\}$. In the following four definitions structures denoted by $\B$ are going to be in the same language $\LL_B = \LL \cup \{U\}$ with $U$ a unary predicate. It is not always necessary to have this predicate but we keep it for the sake of uniformity. Structures denoted by $\A$ will have different languages $\LL_A$ (those are not as important in later applications). -%All the colors $\vec C$ are interpreted by colors in $\TT$ by restriction. +% All the colors $\vec C$ are interpreted by colors in $\TT$ by restriction. \input {vc-trees-all_figures} \begin{Definition} - Fix $c_1 < c_2$ in $T$. Let - \begin{align*} - B &= \{b \in T \mid E_{c_1}(c_2, b) \wedge \neg(b \geq c_2)\} \\ - A &= T - B \\ - S_1 &= \{t \in T \mid t < c_1\} \\ - S_2 &= \{t \in T \mid t < c_2\} \\ - S_B &= S_2 - S_1 \\ - T_A &= \{t \in T \mid c_2 \leq t\} - \end{align*} - Define structures $\A^{c_1}_{c_2} = (A, \leq, \vec C, S_1, T_A)$ and $\B^{c_1}_{c_2} = (B, \leq, \vec C, S_B)$ where $\LL_A$ is an expansion of $\LL$ by two unary predicates (and $\LL_B$ as defined above). Note that $c_1, c_2 \notin B$. + Fix $c_1 < c_2$ in $T$. Let + \begin{align*} + B &= \{b \in T \mid E_{c_1}(c_2, b) \wedge \neg(b \geq c_2)\}, \\ + A &= T - B, \\ + S_1 &= \{t \in T \mid t < c_1\}, \\ + S_2 &= \{t \in T \mid t < c_2\}, \\ + S_B &= S_2 - S_1, \\ + T_A &= \{t \in T \mid c_2 \leq t\}. + \end{align*} + Define structures $\A^{c_1}_{c_2} = (A, \leq, \vec C \cap A, S_1, T_A)$ + where $\vec C \cap A = \curly{C_1 \cap A, \ldots, C_n \cap A}$ + and $\B^{c_1}_{c_2} = (B, \leq, \vec C \cap B, S_B)$ where $\LL_A$ is an expansion of $\LL$ by two unary predicate symbols (and $\LL_B$ as defined above). Note that $c_1, c_2 \notin B$. \end{Definition} \begin{Definition} - Fix $c$ in $T$. Let - \begin{align*} - B &= \{b \in T \mid \neg(b \geq c) \wedge E(b,c)\} \\ - A &= T - B \\ - S_1 &= \{t \in T \mid t < c\} - \end{align*} - Define structures $\A_{c} = (A, \leq, \vec C)$ and $\B_{c} = (B, \leq, \vec C, S_1)$ where $\LL_A = \LL$ (and $\LL_B$ as defined above). Note that $c \notin B$. (cf example \ref{ex_cone}). + Fix $c$ in $T$. Let + \begin{align*} + B &= \{b \in T \mid \neg(b \geq c) \wedge E(b,c)\}, \\ + A &= T - B, \\ + S_1 &= \{t \in T \mid t < c\}. + \end{align*} + Define structures $\A_{c} = (A, \leq, \vec C \cap A)$ and $\B_{c} = (B, \leq, \vec C\cap B, S_1)$ where $\LL_A = \LL$ (and $\LL_B$ as defined above). Note that $c \notin B$. (cf example \ref{ex_cone}). \end{Definition} \begin{Definition} - Fix $c$ in $T$ and $S \subseteq T$ a finite subset. Let - \begin{align*} - B &= \{b \in T \mid (b > c) \text{ and for all $s \in S$ we have } \neg E_c(s, b)\} \\ - A &= T - B \\ - S_1 &= \{t \in T \mid t \leq c\} - \end{align*} - Define structures $\A^{c}_{S} = (A, \leq, \vec C, S_1)$ and $\B^{c}_{S} = (B, \leq, \vec C, B)$ where $L_A$ is an expansion of $\LL$ by a single unary predicate (and $U \in \LL_B$ vacuously interpreted by $B$). Note that $c \notin B$ and $S \cap B = \emptyset$. + Fix $c$ in $T$ and a finite subset $S \subseteq T$. Let + \begin{align*} + B &= \{b \in T \mid (b > c) \text{ and for all $s \in S$ we have } \neg E_c(s, b)\}, \\ + A &= T - B, \\ + S_1 &= \{t \in T \mid t \leq c\}. + \end{align*} + Define structures $\A^{c}_{S} = (A, \leq, \vec C\cap A, S_1)$ and $\B^{c}_{S} = (B, \leq, \vec C\cap B, B)$ where $\LL_A$ is an expansion of $\LL$ by a single unary predicate (and $U \in \LL_B$ vacuously interpreted by $B$). Note that $c \notin B$ and $S \cap B = \emptyset$. \end{Definition} \begin{Definition} - Fix $S \subseteq T$ a finite subset. Let - \begin{align*} - B &= \{b \in T \mid \text{ for all $s \in S$ we have } \neg E(s, b)\} \\ - A &= T - B - \end{align*} - Define structures $\A_{S} = (A, \leq)$ and $\B_{S} = (B, \leq, \vec C, B)$ where $\LL_A = \LL$ (and $U \in \LL_B$ vacuously interpreted by $B$). Note that $S \cap B = \emptyset$. (cf example \ref{ex_disc}) + Fix a finite subset $S \subseteq T$. Let + \begin{align*} + B &= \{b \in T \mid \text{ for all $s \in S$ we have } \neg E(s, b)\}, \\ + A &= T - B. + \end{align*} + Define structures $\A_{S} = (A, \leq)$ and $\B_{S} = (B, \leq, \vec C\cap B, B)$ where $\LL_A = \LL$ (and $U \in \LL_B$ vacuously interpreted by $B$). Note that $S \cap B = \emptyset$. (cf. example \ref{ex_disc}) \end{Definition} \begin{Lemma} \label{subdivide} - The pairs of structures defined above are all proper subdivisions. + The pairs of structures defined above are all proper subdivisions of $\TT$. \end{Lemma} \begin{proof} - We only show this holds for the first definition $\A = \A^{c_1}_{c_2}$ and $\B = \B^{c_1}_{c_2}$. Other cases follow by a similar argument. $A,B$ partition $T$ by definition, so it is a subdivision. To show that it is proper by Lemma \ref{lm_subdivision} we only need to check that it is $0$-proper. Suppose we have - \begin{align*} - a &= (a_1, a_2, \ldots, a_p) \in A^p \\ - a' &= (a_1', a_2', \ldots, a_p') \in A^p \\ - b &= (b_1, b_2, \ldots, b_q) \in B^q \\ - b' &= (b_1', b_2', \ldots, b_q') \in B^q - \end{align*} - with $(\A, a) \equiv_0 (\A, a')$ and $(\B, b) \equiv_0 (\B, b')$. We need to make sure that $ab$ has the same quantifier free type as $a'b'$. Any two elements in $T$ can be related in the four following ways - \begin{align*} - x &= y \\ - x &< y \\ - x &> y \\ - x&,y \text{ are incomparable} - \end{align*} - We need to check that the same relations hold for the pairs of $(a_i, b_j), (a_i', b_j')$ for all $i,j$. - - \begin{itemize} - \item It is impossible that $a_i = b_j$ as they come from disjoint sets. - \item Suppose $a_i < b_j$. This forces $a_i \in S_1$ thus $a_i' \in S_1$ and $a_i' < b_j'$. - \item Suppose $a_i > b_j$ This forces $b_j \in S_B$ and $a \in T_A$, thus $b_j' \in S_B$ and $a_i' \in T_A$ so $a_i' > b_j'$. - \item Suppose $a_i$ and $b_j$ are incomparable. Two cases are possible: - \begin{itemize} - \item $b_j \notin S_B$ and $a_i \in T_A$. Then $b_j' \notin S_B$ and $a_i' \in T_A$ making $a_i', b_j'$ incomparable. - \item $b_j \in S_B$, $a_i \notin T_A$, $a_i \notin S_1$. Similarly this forces $a_i', b_j'$ incomparable. - \end{itemize} - \end{itemize} - Also we need to check that $ab$ has the same colors as $a'b'$. But that is immediate as having the same color in the substructure means having the same color in the whole tree. + We only show this holds for the pair $(\A, \B) = (\A^{c_1}_{c_2} ,\B^{c_1}_{c_2})$. + The other cases follow by a similar argument. + The sets $A,B$ partition $T$ by definition, so $(A,B)$ is a subdivision of $\TT$. + To show that it is proper, by Lemma \ref{lm_subdivision} we only need to check that it is $0$-proper. Suppose we have + \begin{align*} + a &= (a_1, a_2, \ldots, a_p) \in A^p, \\ + a' &= (a_1', a_2', \ldots, a_p') \in A^p, \\ + b &= (b_1, b_2, \ldots, b_q) \in B^q, \\ + b' &= (b_1', b_2', \ldots, b_q') \in B^q. + \end{align*} + with $\A \models a \equiv_0 a'$ and $\B \models b \equiv_0 b'$. + We need to show that $ab$ has the same quantifier-free type in $\TT$ as $a'b'$. + Any two elements in $T$ can be related in the four following ways: + \begin{align*} + x &= y, \\ + x &< y, \\ + x &> y, \text{ or } \\ + x&,y \text{ are incomparable.} + \end{align*} + We need to check that for all $i,j$ the same relations hold for $(a_i, b_j)$ as do for $(a_i', b_j')$. + + \begin{itemize} + \item It is impossible that $a_i = b_j$ as they come from disjoint sets. + \item Suppose $a_i < b_j$. This forces $a_i \in S_1$ thus $a_i' \in S_1$ and $a_i' < b_j'$. + \item Suppose $a_i > b_j$. This forces $b_j \in S_B$ and $a \in T_A$, thus $b_j' \in S_B$ and $a_i' \in T_A$, so $a_i' > b_j'$. + \item Suppose $a_i$ and $b_j$ are incomparable. Two cases are possible: + \begin{itemize} + \item $b_j \notin S_B$ and $a_i \in T_A$. Then $b_j' \notin S_B$ and $a_i' \in T_A$ making $a_i', b_j'$ incomparable. + \item $b_j \in S_B$, $a_i \notin T_A$, $a_i \notin S_1$. Similarly this forces $a_i', b_j'$ to be incomparable. + \end{itemize} + \end{itemize} + Also we need to check that $ab$ has the same colors as $a'b'$. But that is immediate as having the same color in a substructure means having the same color in the tree. \end{proof} \section{Main proof} -Basic idea for the proof is as follows. Suppose we have a formula with $q$ parameters. We are able to split our parameter space into $O(n)$ many partitions. Each of $q$ parameters can come from any of those $O(n)$ partitions giving us $O(n)^q$ many choices for parameter configuration. When every parameter is coming from a fixed partition the number of definable sets is constant and in fact is uniformly bounded above by some $N$. This gives us at most $N \cdot O(n)^q$ possibilities for different definable sets. +The basic idea for the proof is as follows. +Suppose we have a formula with $q$ parameters over a parameter set of size $n$. +We are able to split our parameter space into $O(n)$ many partitions. Each of $q$ parameters can come from any of those $O(n)$ partitions giving us $O(n)^q$ many choices for parameter configuration. When every parameter is coming from a fixed partition the number of definable sets is constant and in fact is uniformly bounded above by some $N$. This gives us at most $N \cdot O(n)^q$ possibilities for different definable sets. First, we generalize Corollary \ref{cor_type_count}. (This is required for computing vc-density for formulas $\phi(x, y)$ with $|y| > 1$). \begin{Lemma} \label{lm_partition_bound} - Consider a finite collection $(\A_i, \B_i)_{i \leq n}$ where each $(\A_i, \B_i)$ is either a proper subdivision or a singleton: $B_i = \{b_i\}$ with $A_i = T$. Also assume that all $\B_i$ have the same language $\LL_B$. Let $A = \bigcap_{i \in I} A_i$. Fix a formula $\phi(x, y)$ of complexity $m$ . Let $N = N(m, |y|, \LL_B)$ as in Definition \ref{def_type_count}. Consider any $B \subseteq T^{|y|}$ of the form - \begin{align*} - B = B_1^{i_1} \times B_2^{i_2} \times \ldots \times B_n^{i_n} \text { with } i_1 + i_2 + \ldots + i_n = |y| - \end{align*} - (some of the indexes can be zero). Then we have the following bound - \begin{align*} - \phi(A^{|x|}, B) \leq N^{|y|} - \end{align*} + Consider a finite collection $(\A_i, \B_i)_{i \leq n}$ satisfying the following properties: + \begin{itemize} + \item $(\A_i, \B_i)$ is either a proper subdivision of $\TT$ or $A_i = T$ and $B_i = \{b_i\}$, + \item all $\B_i$ have the same language $\LL_B$, + \item sets $\curly{B_i}_{i \leq n}$ are pairwise disjoint. + \end{itemize} + Let $A = \bigcap_{i \in I} A_i$. + Fix a formula $\phi(x, y)$ of complexity $m$ . Let $N = N(m, |y|, \LL_B)$ be as in Definition \ref{def_type_count}. Consider any $B \subseteq T^{|y|}$ of the form + \begin{align*} + B = B_1^{i_1} \times B_2^{i_2} \times \ldots \times B_n^{i_n} \text { with } i_1 + i_2 + \ldots + i_n = |y|. + \end{align*} + (some of the indeces can be zero). Then we have the following bound: + \begin{align*} + \phi(A^{|x|}, B) \leq N^{|y|}. + \end{align*} \end{Lemma} \begin{proof} - We show this result by counting types. Suppose we have - \begin{align*} - b_1, b_1' &\in B_1^{i_1} \text{ with } b_1 \equiv_m b_1' \text { in } \B_1 \\ - b_2, b_2' &\in B_2^{i_2} \text{ with } b_2 \equiv_m b_2' \text { in } \B_2 \\ - &\cdots \\ - b_n, b_n' &\in B_n^{i_n} \text{ with } b_n \equiv_m b_n' \text { in } \B_n - \end{align*} - Then we have - \begin{align*} - \phi(A^{|x|}, b_1, b_2, \ldots b_n) \ifff \phi(A^{|x|}, b_1', b_2', \ldots b_n') - \end{align*} - This is easy to see by applying Corollary \ref{cor_type_count} one by one for each tuple. This works if $\B_i$ is a part of a proper subdivision; if it is a singleton then the implication is trivial as $b_i = b_i'$. - Thus $\phi(A^{|x|}, B)$ only depends on the choice of the types for the tuples - \begin{align*} - |\phi(A^{|x|}, B)| \leq |S^m_{\B_1, i_1}| \cdot |S^m_{\B_2, i_2}| \cdot \ldots \cdot |S^m_{\B_n, i_n}| - \end{align*} - Now for each type space we have an inequality - \begin{align*} - |S^m_{\B_j, i_j}| \leq N(m, i_j, \LL_B) \leq N(m, |y|, \LL_B) \leq N - \end{align*} - (For singletons $|S^m_{\B_j, i_j}| = 1 \leq N$). Only non-zero indexes contribute to the product and there are at most $|y|$ of those (by equality $i_1 + i_2 + \ldots + i_n = |y|$). Thus we have - \begin{align*} - |\phi(A^{|x|}, B)| \leq N^{|y|} - \end{align*} - as needed. + We show this result by counting types. + \begin{Claim} + Suppose we have + \begin{align*} + b_1, b_1' &\in B_1^{i_1} \text{ with } b_1 \equiv_m b_1' \text { in } \B_1, \\ + b_2, b_2' &\in B_2^{i_2} \text{ with } b_2 \equiv_m b_2' \text { in } \B_2, \\ + &\cdots \\ + b_n, b_n' &\in B_n^{i_n} \text{ with } b_n \equiv_m b_n' \text { in } \B_n. + \end{align*} + Then + \begin{align*} + \phi(A^{|x|}, b_1, b_2, \ldots b_n) \iff \phi(A^{|x|}, b_1', b_2', \ldots b_n'). + \end{align*} + \end{Claim} + \begin{proof} + Define $\bar b_i = (b_1, \ldots, b_i, b_{i+1}', \ldots, b_n') \in B$ for $i \in [0..n]$. + (That is, a tuple where first $i$ elements are without prime, and elements after that are with a prime.) + We have $\phi(A^{|x|}, \bar b_i) \iff \phi(A^{|x|}, \bar b_{i+1})$ as either $(\A_{i+1}, \B_{i+1})$ is $m$-proper + or $\B_{i+1}$ is a singleton, and the implication is trivial. + (Notice that $b_i \in \A_j$ for $j \neq i$ by disjointness assumption.) + Thus, by induction we get $\phi(A^{|x|}, \bar b_0) \iff \phi(A^{|x|}, \bar b_n)$ as needed. + \end{proof} + Thus $\phi(A^{|x|}, B)$ only depends on the choice of the types for the tuples: + \begin{align*} + |\phi(A^{|x|}, B)| \leq |S^m_{\B_1, i_1}| \cdot |S^m_{\B_2, i_2}| \cdot \ldots \cdot |S^m_{\B_n, i_n}| + \end{align*} + Now for each type space we have an inequality + \begin{align*} + |S^m_{\B_j, i_j}| \leq N(m, i_j, \LL_B) \leq N(m, |y|, \LL_B) \leq N + \end{align*} + (For singletons $|S^m_{\B_j, i_j}| = 1 \leq N$). Only non-zero indeces contribute to the product and there are at most $|y|$ of those (by the equality $i_1 + i_2 + \ldots + i_n = |y|$). Thus we have + \begin{align*} + |\phi(A^{|x|}, B)| \leq N^{|y|} + \end{align*} + as needed. \end{proof} For subdivisions to work out properly, we will need to work with subsets closed under meets. We observe that the closure under meets doesn't add too many new elements. -%MAYBE: write a more detailed proof +% MAYBE: write a more detailed proof \begin{Lemma} \label{lm_meet} - Suppose $S \subseteq T$ is a finite subset of size $n$ in a meet tree and $S'$ is its closure under meets. Then $|S'| \leq 2n$. + Suppose $S \subseteq T$ is a finite subset of size $n \geq 1$ in a meet tree and $S'$ is its closure under meets. Then $|S'| \leq 2n - 1$. \end{Lemma} \begin{proof} - We can partition $S$ into connected components and prove the result separately for each component. Thus we may assume elements of $S$ lie in the same connected component. We prove the claim by induction on $n$. Base case $n = 1$ is clear. Suppose we have $S$ of size $k$ with closure of size at most $2k - 1$. Take a new point $s$, and look at its meets with all the elements of $S$. Pick the smallest one, $s'$. Then $S \cup \{s, s'\}$ is closed under meets. + We can partition $S$ into connected components and prove the result separately for each component. Thus we may assume all elements of $S$ lie in the same connected component. We prove the claim by induction on $n$. The base case $n = 1$ is clear. Suppose we have $S$ of size $k$ with closure of size at most $2k - 1$. Take a new point $s$, and look at its meets with all the elements of $S$. Pick the smallest one, $s'$. Then $S \cup \{s, s'\}$ is closed under meets. \end{proof} Putting all of those results together we are able to compute the $\vc$-density of formulas in meet trees. \begin{Theorem} - Let $\TT$ be an infinite (colored) meet tree and $\phi(x, y)$ a formula with $|x| = p$ and $|y| = q$. Then $\vc(\phi) \leq q$. + Let $\TT$ be an infinite (colored) meet tree and $\phi(x, y)$ a formula with $|x| = p$ and $|y| = q$. Then $\vc(\phi) \leq q$. \end{Theorem} \begin{proof} - Pick a finite subset of $S_0 \subset T^p$ of size $n$. Let $S_1 \subset T$ consist of coordinates of $S_0$. Let $S \subset T$ be a closure of $S_1$ under meets. Using Lemma \ref{lm_meet} we have $|S| \leq 2|S_1| \leq 2p|S_0| = 2pn = O(n)$. We have $S_0 \subseteq S^p$, so $|\phi(S_0, T^q)| \leq |\phi(S^p, T^q)|$. Thus it is enough to show $|\phi(S^p, T^q)| = O(n^q)$. - - Label $S = \{c_i\}_{i \in I}$ with $|I| \leq 2pn$. For every $c_i$ we construct two partitions in the following way. We have that $c_i$ is either minimal in $S$ or it has a predecessor in $S$ (greatest element less than $c$). If it is minimal, construct $(\A_{c_i}, \B_{c_i})$. If there is a predecessor $p$, construct $(\A^p_{c_i}, \B^p_{c_i})$. For the second subdivision let $G$ be all the elements in $S$ greater than $c_i$ and construct $(\A^c_G, \B^c_G)$. So far we have constructed two subdivisions for every $i \in I$. Additionally construct $(\A_S, \B_S)$. We end up with a finite collection of proper subdivisons $(\A_j, \B_j)_{j \in J}$ with $|J| = 2|I| + 1$. Before we proceed, we note the following two lemmas describing our partitions. - - \begin{Lemma} - For all $j \in J$ we have $S \subseteq A_j$. Thus $S \subseteq \bigcap_{j \in J} A_j$ and $S^p \subseteq \bigcap_{j \in J} (A_j)^p$. - \end{Lemma} - - \begin{proof} - Check this for each possible choice of partition. Cases for partitions of the type $\A_S, \A^c_G, \A_c$ are easy. Suppose we have a partition $(\A, \B) = (\A^{c_1}_{c_2}, \B^{c_1}_{c_2})$. We need to show that $B \cap S = \emptyset$. By construction we have $c_1, c_2 \notin B$. Suppose we have some other $c \in S$ with $c \in B$. We have $E_{c_1}(c_2, c)$ i.e. there is some $b$ such that $(b > c_1)$, $(b \leq c_2)$ and $(b \leq c)$. Consider the meet $(c \wedge c_2)$. We have $(c \wedge c_2) \geq b > c_1$. Also as $\neg (c \geq c_2)$ we have $(c \wedge c_2) < c_2$. To summarize: $c_2 > (c \wedge c_2) > c_1$. But this contradicts our construction as $S$ is closed under meets, so $(c \wedge c_2) \in S$ and $c_1$ is supposed to be a predecessor of $c_2$ in $S$. - \end{proof} - - \begin{Lemma} - $\{B_j\}_{j \in J}$ is a disjoint partition of $T - S$ i.e. $T = \bigsqcup_{j \in J} B_j \sqcup S$ - \end{Lemma} - - \begin{proof} - This more or less follows from the choice of partitions. Pick any $b \in S - T$. Take all the elements in $S$ greater than $b$ and take the minimal one $a$. Take all the elements in $S$ less than $b$ and take the maximal one $c$ (possible as $S$ is closed under meets). Also take all the elements in $S$ incomparable to $b$ and denote them $G$. If both $a$ and $c$ exist we have $b \in \B^a_c$. If only the upper bound exists we have $b \in \B^a_G$. If only the lower bound exists we have $b \in \B_c$. If neither exists we have $b \in \B_G$. - \end{proof} - - \begin{Note} - Those two lemmas imply $S = \bigcap_{j \in J} A_j$. - \end{Note} - - \begin{Note} - %careful application of note - have different languages and has to be > 1 - For one-dimensional case $q = 1$ we don't need to do any more work. We have partitioned the parameter space into $|J| = O(n)$ many pieces and over each piece the number of definable sets is uniformly bounded. By Corollary \ref{cor_type_count} we have that $|\phi((A_j)^p, B_j)| \leq N$ for any $j \in J$ (letting $N = N(n_\phi, q, \LL \cup \{S\})$ where $n_\phi$ is the complexity of $\phi$ and $S$ is a unary predicate). Compute - %describe steps - \begin{align*} - |\phi(S^p, T)| - &= \left|\bigcup_{j \in J} \phi(S^p, B_j) \cup \phi(S^p, S)\right| \leq \\ - &\leq \sum_{j \in J} |\phi(S^p, B_j)| + |\phi(S^p, S)| \leq \\ - &\leq \sum_{j \in J} |\phi((A_j)^p, B_j)| + |S| \leq \\ - &\leq \sum_{j \in J}N + |I| \leq \\ - &\leq (4pn + 1)N + 2pn = (4pN + 2p)n + N = O(n) - \end{align*} - \end{Note} - Basic idea for the general case $q \geq 1$ is that we have $q$ parameters and $|J| = O(n)$ many partitions to pick each parameter from giving us $|J|^q = O(n^q)$ choices for the parameter configuration, each giving a uniformly constant number of definable subsets of $S$. (If every parameter is picked from a fixed partition, Lemma \ref{lm_partition_bound} provides a uniform bound). This yields $\vc(\phi) \leq q$ as needed. The rest of the proof is stating this idea formally. - - First, we extend our collection of subdivisions $(\A_j, \B_j)_{j \in J}$ by the following singleton sets. For each $c_i \in S$ let $B_i = \{c_i\}$ and $A_i = T$ and add $(\A_i, \B_i)$ to our collection with $\LL_B$ the language of $B_i$ interpreted arbitrarily. We end up with a new collection $(\A_k, \B_k)_{k \in K}$ indexed by some $K$ with $|K| = |J| + |I|$ (we added $|S|$ new pairs). Now ${B_k}_{k \in K}$ partitions $T$, so $T = \bigsqcup_{k \in K} B_k$ and $S = \bigcap_{j \in J} A_j = \bigcap_{k \in K} A_k$. For $(k_1, k_2, \ldots k_q) = \vec k \in K^q$ denote - \begin{align*} - B_{\vec k} = B_{k_1} \times B_{k_2} \times \ldots \times B_{k_q} - \end{align*} - Then we have the following identity - \begin{align*} - T^q = (\bigsqcup_{k \in K} B_k)^q = \bigsqcup_{\vec k \in K^q} B_{\vec k} - \end{align*} - Thus we have that $\{B_{\vec k}\}_{\vec k \in K^q}$ partition $T^q$. Compute - \begin{align*} - |\phi(S^p, T^q)| - &= \left|\bigcup_{\vec k \in K^q} \phi(S^p, B_{\vec k}) \right| \leq \\ - &\leq \sum_{\vec k \in K^q} |\phi(S^p, B_{\vec k})| - \end{align*} - We can bound $|\phi(S^p, B_{\vec k})|$ uniformly using Lemma \ref{lm_partition_bound}. $(\A_k, \B_k)_{k \in K}$ satisfies the requirements of the lemma and $B_{\vec k}$ looks like $B$ in the lemma after possibly permuting some variables in $\phi$. Applying the lemma we get - \begin{align*} - |\phi(S^p, B_{\vec k})| \leq N^q - \end{align*} - with $N$ only depending on $q$ and complexity of $\phi$. We complete our computation - \begin{align*} - |\phi(S^p, T^q)| - &\leq \sum_{\vec k \in K^q} |\phi(S^p, B_{\vec k})| \leq \\ - &\leq \sum_{\vec k \in K^q} N^q \leq \\ - &\leq |K^q| N^q \leq \\ - &\leq (|J| + |I|)^q N^q \leq \\ - &\leq (4pn + 1 + 2pn)^q N^q = N^q (6p + 1/n)^q n^q = O(n^q) - \end{align*} - \end{proof} - \begin{Corollary} - In the theory of infinite (colored) meet trees we have $vc(n) = n$ for all $n$. - \end{Corollary} - We get the general result for the trees that aren't necessarily meet trees via an easy application of interpretability. - \begin{Corollary} - In the theory of infinite (colored) trees we have $vc(n) = n$ for all $n$. - \end{Corollary} - \begin{proof} - Let $\TT'$ be a tree. We can embed it in a larger tree that is closed under meets $\TT' \subset \TT$. Expand $\TT$ by an extra color and interpret it by coloring the subset $\TT'$. Thus we can interpret $\TT'$ in $T^1$. By Corollary 3.17 in \cite{vc_density} we get that $\vc^{\TT'}(n) \leq \vc^T(1 \cdot n) = n$ thus $\vc^{\TT'}(n) = n$ as well. - \end{proof} - - - - \begin{thebibliography}{9} - -\bibitem{vc_density} + Pick a finite subset of $S_0 \subset T^p$ of size $n$. Let $S_1 \subset T$ consist of the components of the elements of $S_0$. Let $S \subset T$ be the closure of $S_1$ under meets. Using Lemma \ref{lm_meet} we have $|S| \leq 2|S_1| \leq 2p|S_0| = 2pn = O(n)$. We have $S_0 \subseteq S^p$, so $|\phi(S_0, T^q)| \leq |\phi(S^p, T^q)|$. Thus it is enough to show $|\phi(S^p, T^q)| = O(n^q)$. + + Label $S = \{c_i\}_{i \in I}$ with $|I| \leq 2pn$. For every $c_i$ we construct two partitions in the following way. We have that $c_i$ is either minimal in $S$ or it has a predecessor in $S$ (greatest element less than $c$). If it is minimal, construct $(\A_{c_i}, \B_{c_i})$. If there is a predecessor $p$, construct $(\A^p_{c_i}, \B^p_{c_i})$. For the second subdivision let $G$ be all the elements in $S$ greater than $c_i$ and construct $(\A^c_G, \B^c_G)$. So far we have constructed two subdivisions for every $i \in I$. Additionally construct $(\A_S, \B_S)$. We end up with a finite collection of proper subdivisons $(\A_j, \B_j)_{j \in J}$ with $|J| = 2|I| + 1$. Before we proceed, we note the following two lemmas describing our partitions. + + \begin{Lemma} + For all $j \in J$ we have $S \subseteq A_j$. Thus $S \subseteq \bigcap_{j \in J} A_j$ and $S^p \subseteq \bigcap_{j \in J} (A_j)^p$. + \end{Lemma} + + \begin{proof} + Check this for each possible choice of partition. Cases for partitions of the type $\A_S, \A^c_G, \A_c$ are easy. Suppose we have a partition $(\A, \B) = (\A^{c_1}_{c_2}, \B^{c_1}_{c_2})$. We need to show that $B \cap S = \emptyset$. By construction we have $c_1, c_2 \notin B$. Suppose we have some other $c \in S$ with $c \in B$. We have $E_{c_1}(c_2, c)$ i.e. there is some $b$ such that $(b > c_1)$, $(b \leq c_2)$ and $(b \leq c)$. Consider the meet $(c \wedge c_2)$. We have $(c \wedge c_2) \geq b > c_1$. Also as $\neg (c \geq c_2)$ we have $(c \wedge c_2) < c_2$. To summarize: $c_2 > (c \wedge c_2) > c_1$. But this contradicts our construction as $S$ is closed under meets, so $(c \wedge c_2) \in S$ and $c_1$ is supposed to be a predecessor of $c_2$ in $S$. + \end{proof} + + \begin{Lemma} + $\{B_j\}_{j \in J}$ is a disjoint partition of $T - S$ i.e. $T = \bigsqcup_{j \in J} B_j \sqcup S$ + \end{Lemma} + + \begin{proof} + This more or less follows from the choice of partitions. Pick any $b \in S - T$. Take all the elements in $S$ greater than $b$ and take the minimal one $a$. Take all the elements in $S$ less than $b$ and take the maximal one $c$ (possible as $S$ is closed under meets). Also take all the elements in $S$ incomparable to $b$ and denote them $G$. If both $a$ and $c$ exist we have $b \in \B^a_c$. If only the upper bound exists we have $b \in \B^a_G$. If only the lower bound exists we have $b \in \B_c$. If neither exists we have $b \in \B_G$. + \end{proof} + + \begin{Note} + Those two lemmas imply $S = \bigcap_{j \in J} A_j$. + \end{Note} + + \begin{Note} + % careful application of note - have different languages and has to be > 1 + For one-dimensional case $q = 1$ we don't need to do any more work. We have partitioned the parameter space into $|J| = O(n)$ many pieces and over each piece the number of definable sets is uniformly bounded. By Corollary \ref{cor_type_count} we have that $|\phi((A_j)^p, B_j)| \leq N$ for any $j \in J$ (letting $N = N(n_\phi, q, \LL \cup \{S\})$ where $n_\phi$ is the complexity of $\phi$ and $S$ is a unary predicate). Compute + % describe steps + \begin{align*} + |\phi(S^p, T)| + &= \left|\bigcup_{j \in J} \phi(S^p, B_j) \cup \phi(S^p, S)\right| \leq \\ + &\leq \sum_{j \in J} |\phi(S^p, B_j)| + |\phi(S^p, S)| \leq \\ + &\leq \sum_{j \in J} |\phi((A_j)^p, B_j)| + |S| \leq \\ + &\leq \sum_{j \in J}N + |I| \leq \\ + &\leq (4pn + 1)N + 2pn = (4pN + 2p)n + N = O(n) + \end{align*} + \end{Note} + Basic idea for the general case $q \geq 1$ is that we have $q$ parameters and $|J| = O(n)$ many partitions to pick each parameter from giving us $|J|^q = O(n^q)$ choices for the parameter configuration, each giving a uniformly constant number of definable subsets of $S$. (If every parameter is picked from a fixed partition, Lemma \ref{lm_partition_bound} provides a uniform bound). This yields $\vc(\phi) \leq q$ as needed. The rest of the proof is stating this idea formally. + + First, we extend our collection of subdivisions $(\A_j, \B_j)_{j \in J}$ by the following singleton sets. For each $c_i \in S$ let $B_i = \{c_i\}$ and $A_i = T$ and add $(\A_i, \B_i)$ to our collection with $\LL_B$ the language of $B_i$ interpreted arbitrarily. We end up with a new collection $(\A_k, \B_k)_{k \in K}$ indexed by some $K$ with $|K| = |J| + |I|$ (we added $|S|$ new pairs). Now $\curly{B_k}_{k \in K}$ partitions $T$, so $T = \bigsqcup_{k \in K} B_k$ and $S = \bigcap_{j \in J} A_j = \bigcap_{k \in K} A_k$. For $(k_1, k_2, \ldots k_q) = \vec k \in K^q$ denote + \begin{align*} + B_{\vec k} = B_{k_1} \times B_{k_2} \times \ldots \times B_{k_q} + \end{align*} + Then we have the following identity + \begin{align*} + T^q = (\bigsqcup_{k \in K} B_k)^q = \bigsqcup_{\vec k \in K^q} B_{\vec k} + \end{align*} + Thus we have that $\{B_{\vec k}\}_{\vec k \in K^q}$ partition $T^q$. Compute + \begin{align*} + |\phi(S^p, T^q)| + &= \left|\bigcup_{\vec k \in K^q} \phi(S^p, B_{\vec k}) \right| \leq \\ + &\leq \sum_{\vec k \in K^q} |\phi(S^p, B_{\vec k})| + \end{align*} + We can bound $|\phi(S^p, B_{\vec k})|$ uniformly using Lemma \ref{lm_partition_bound}. $(\A_k, \B_k)_{k \in K}$ satisfies the requirements of the lemma and $B_{\vec k}$ looks like $B$ in the lemma after possibly permuting some variables in $\phi$. Applying the lemma we get + \begin{align*} + |\phi(S^p, B_{\vec k})| \leq N^q + \end{align*} + with $N$ only depending on $q$ and complexity of $\phi$. We complete our computation + \begin{align*} + |\phi(S^p, T^q)| + &\leq \sum_{\vec k \in K^q} |\phi(S^p, B_{\vec k})| \leq \\ + &\leq \sum_{\vec k \in K^q} N^q \leq \\ + &\leq |K^q| N^q \leq \\ + &\leq (|J| + |I|)^q N^q \leq \\ + &\leq (4pn + 1 + 2pn)^q N^q = N^q (6p + 1/n)^q n^q = O(n^q) + \end{align*} +\end{proof} +\begin{Corollary} + In the theory of infinite (colored) meet trees we have $vc(n) = n$ for all $n$. +\end{Corollary} +We get the general result for the trees that aren't necessarily meet trees via an easy application of interpretability. +\begin{Corollary} + In the theory of infinite (colored) trees we have $vc(n) = n$ for all $n$. +\end{Corollary} +\begin{proof} + Let $\TT'$ be a tree. We can embed it in a larger tree $\TT$ that is closed under meets. Expand $\TT$ by an extra color and interpret it by coloring the subset $\TT'$. Thus we can interpret $\TT'$ in $T$. By Corollary 3.17 in \cite{density} we get that $\vc^{\TT'}(n) \leq \vc^T(1 \cdot n) = n$ thus $\vc^{\TT'}(n) = n$ as well. +\end{proof} + +This settles the question of $vc$-function for trees. Lacking a quantifier elimination result, a lot is still not known. +One can try to adapt these techniques to compute vc-density of a fixed formula, and see if it can achieve any interesting non-integer values. +It is also not known whether trees have VC 1 property (see \cite{density} 5.2 for definition). +Our technique can show that VC 2 property holds, but it wouldn't give an optimal vc-function. + +One can also try to apply similar techniques to more general classes of patrially ordered sets. +For example, it is not known how vc-density behaves in lattices. +Similarly, dropping the order, one can look a nicely behaved families of graphs, such as planar graphs or flat graphs. +Those are known to be dp-minimal, so one would expect a simple vc-function. +It is this author's hope that the techniques developed in this paper can be adapted to yield fruitful results for a more general class of structures. + +\begin{thebibliography}{9} + +\bibitem{density} M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko, \textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, Trans. Amer. Math. Soc. 368 (2016), 5889-5949 \bibitem{simon_dp_min} - P. Simon, - \textit{On dp-minimal ordered structures}, - J. Symbolic Logic 76 (2011), no. 2, 448-460 + P. Simon, + \textit{On dp-minimal ordered structures}, + J. Symbolic Logic 76 (2011), no. 2, 448-460 \bibitem{parigot_trees} - Michel Parigot. - Th\'eories d'arbres. - \textit{Journal of Symbolic Logic}, 47, 1982, 841-853 - - + Michel Parigot. + Th\'eories d'arbres. + \textit{Journal of Symbolic Logic}, 47, 1982, 841-853 + + \end{thebibliography} \end{document} diff --git a/research/02 Trees vc-density/vc-trees-all_figures.tex b/research/02 Trees vc-density/vc-trees-all_figures.tex index 7f80340f..e5edd1ac 100644 --- a/research/02 Trees vc-density/vc-trees-all_figures.tex +++ b/research/02 Trees vc-density/vc-trees-all_figures.tex @@ -40,7 +40,7 @@ \begin{figure}[p] \input {vc-trees-fig_1} - \caption{Proper subdivision for $(A, B) = (A^{c_1}_G, B^{c_1}_G)$ for $G = \{c_2\}$} + \caption{Proper subdivision for $(A, B) = (A^{c_1}_G, B^{c_1}_G)$ for $S = \{c_2\}$} \end{figure} \tikzstyle{up}=[node, fill = white] @@ -54,5 +54,5 @@ \begin{figure}[p] \input {vc-trees-fig_1} - \caption{Proper subdivision for $(A, B) = (A_G, B_G)$ for $G = \{c_1, c_2\}$} + \caption{Proper subdivision for $(A, B) = (A_G, B_G)$ for $S = \{c_1, c_2\}$} \end{figure} diff --git a/research/08 shelah-spencer VC/shelah_spencer.pdf b/research/08 shelah-spencer VC/shelah_spencer.pdf index 052e5223..caa8a4b5 100644 Binary files a/research/08 shelah-spencer VC/shelah_spencer.pdf and b/research/08 shelah-spencer VC/shelah_spencer.pdf differ diff --git a/research/08 shelah-spencer VC/shelah_spencer.tex b/research/08 shelah-spencer VC/shelah_spencer.tex index 895a1c1d..aa2b0c88 100644 --- a/research/08 shelah-spencer VC/shelah_spencer.tex +++ b/research/08 shelah-spencer VC/shelah_spencer.tex @@ -65,7 +65,7 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -VC-density was studied in \cite{vc_density} by Aschenbrenner, Dolich, Haskell, MacPherson, and Starchenko as a natural notion of dimension for NIP theories. In an NIP theory we can define a vc-function +VC-density was studied in \cite{density} by Aschenbrenner, Dolich, Haskell, MacPherson, and Starchenko as a natural notion of dimension for NIP theories. In an NIP theory we can define a vc-function \begin{align*} \vc : \N \arr \N @@ -90,7 +90,7 @@ where $K(\phi), Y(\phi), \epsilon(\phi)$ are paramters easily computable from the quantifier free form of $\phi$. Chapter 1 introduces basic facts about VC-dimension and vc-density. -More can be found in \cite{vc_density}. +More can be found in \cite{density}. Chapter 2 summarizes notation and basic facts concerning Shelah-Spencer graphs. We direct the reader to \cite{laskowski} for a more in-depth treatment. In chapter 3 we introduce some measure of dimension for quantifier free formulas as well as proving some elementary facts about it. @@ -101,155 +101,7 @@ % Our ambient model is $\MM$. % Notations we use are $\delta(\A), \delta(\A/\B), \A \leq \B$ as well as notions of $N$-strong substructure, minimal extension, chain minimal extension, minimal pair, and $N$-strong closure. -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\section{VC-dimension and vc-density} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\begin{Definition} - Throughout this section we work with a collection $\F$ of subsets of a set $X$. - We call the pair $(X, \F)$ a \defn{set system}. - \begin{itemize} - \item Given a subset $A$ of $X$, we define the set system $(A, A \cap \F)$ - where $A \cap \F = \curly{A \cap F}_{F\in \F}$. - \item For $A \subset X$ we say that $\F$ \defn{shatters} $A$ if $A \cap \F = \PP(A)$. - \end{itemize} -\end{Definition} - -\begin{Definition} - We say $(X, \F)$ has VC-dimension $n$ if the largest subset of $X$ shattered by $\F$ is of size $n$. - If $\F$ shatters arbitrarily large subsets of $X$, we say that $(X, \F)$ has infinite VC-dimension. - We denote the VC-dimension of $(X, \F)$ by $\VC(\F)$. -\end{Definition} - -\begin{Note} - We may drop $X$ from the previous definition, as the VC-dimension doesn't depend on the base set and is determined by $(\bigcup \F, \F)$. -\end{Note} -This allows us to distinguish between well-behaved set systems of finite VC-dimension which tend to have good combinatorial properties and -poorly behaved set systems with infinite VC-dimension. - -Another natural combinatorial notion is that of a dual system: -\begin{Definition} - For $a \in X$ define $X_a = \curly{F \in \F \mid a \in F}$. - Let $\F^* = \curly{X_a}_{a \in X}$. - We define $(\F, \F^*)$ as the \defn{dual system} of $(X, \F)$. - The VC-dimension of the dual system of $(X, \F)$ is referred to as the \defn{dual VC-dimension} of $(X, \F)$ and denoted by $\VC^*(\F)$. - (As before, this notion doesn't depend on $X$.) -\end{Definition} - -\begin{Lemma} - A set system has finite VC-dimension if and only if its dual system has finite VC-dimension. - More precisely - \begin{align*} - \VC^*(\F) \leq 2^{1+\VC(\F)}. - \end{align*} -\end{Lemma} - -For a more refined notion we look at the traces of our family on finite sets: -\begin{Definition} - Define the \defn{shatter function} $\pi_\F \colon \N \arr \N$ and the \defn{dual shatter function} $\pi^*_\F \colon \N \arr \N$ of $\F$ by - \begin{align*} - \pi_\F(n) &= \max \curly{|A \cap \F| \mid A \subset X \text{ and } |A| = n} \\ - \pi^*_\F(n) &= \max \curly{\text{atoms($B$)} \mid B \subset \F, |B| = n} - \end{align*} - where atoms($B$) = number of atoms in the Boolean algebra generated by $B$. - Note that the dual shatter function is precisely the shatter function of the dual system: $\pi^*_\F = \pi_{\F^*}$. -\end{Definition} - -A simple upper bound is $\pi_\F(n) \leq 2^n$ (same for the dual). -If VC-dimension is infinite then clearly $\pi_\F(n) = 2^n$ for all $n$. Conversely we have the following remarkable fact: -\begin{Theorem} [Sauer-Shelah '72] - If the set system $(X, \F)$ has finite VC-dimension $d$ then $\pi_\F(n) \leq {n \choose \leq d}$ where - ${n \choose \leq d} = {n \choose d} + {n \choose d - 1} + \ldots + {n \choose 1}$. -\end{Theorem} - -Thus the systems with a finite VC-dimension are precisely the systems where the shatter function grows polynomially. -Define vc-density to be the degree of that polynomial: -\begin{Definition} - Define \defn{vc-density} and \defn{dual vc-density} of $\F$ as - \begin{align*} - \vc(\F) &= \limsup_{n \to \infty}\frac{\log \pi_\F(n)}{\log n} \in \R^{\geq 0} \cup \curly{+\infty}\\ - \vc^*(\F) &= \limsup_{n \to \infty}\frac{\log \pi^*_\F(n)}{\log n}\in \R^{\geq 0} \cup \curly{+\infty} - \end{align*} -\end{Definition} - -Generally speaking a shatter function that is bounded by a polynomial doesn't itself have to be a polynomial. -Proposition 4.12 in \cite{vc_density} gives an example of a shatter function that grows like $n \log n$ (so it has vc-density $1$). - -So far the notions that we have defined are purely combinatorial. -We now adapt VC-dimension and vc-density to the model theoretic context. - -\begin{Definition} - Work in a structure $M$. - Fix a finite collection of formulas $\Phi(x, y) = \curly{\phi_i(x, y)}$. - - \begin{itemize} - \item For $\phi(x, y) \in \LL(M)$ and $b \in M^{|y|}$ let - \begin{align*} - \phi(M^{|x|}, b) = \{a \in M^{|x|} \mid \phi(a, b)\} \subseteq M^{|x|}. - \end{align*} - \item Let $\Phi(M^{|x|}, M^{|y|})= \{\phi_i(M^{|x|}, b) \mid \phi_i \in \Phi, b \in M^{|y|}\} \subseteq \PP(M^{|x|})$. - \item Let $\F_\Phi = \Phi(M^{|x|}, M^{|y|})$ giving a set system $(M^{|x|}, \F_\Phi)$. - \item Define \defn{VC-dimension} of $\Phi$, $\VC(\Phi)$ to be the VC-dimension of $(M^{|x|}, \F_\Phi)$, similarly for the dual. - \item Define \defn{vc-density} of $\Phi$, $\vc(\Phi)$ to be the vc-density of $(M^{|x|}, \F_\Phi)$, similarly for the dual. - \end{itemize} - - We will also refer to the vc-density and VC-dimension of a single formula $\phi$ - viewing it as a one element collection $\curly{\phi}$. -\end{Definition} - -Counting atoms of a Boolean algebra in a model theoretic setting corresponds to counting types, -so it is instructive to rewrite the shatter function in terms of types. - -\begin{Definition} - \begin{align*} - \pi^*_\Phi(n) &= \max \curly{\text{number of $\Phi$-types over $B$} \mid B \subset M, |B| = n} - \end{align*} -\end{Definition} - -\begin{Lemma} \label{count_types} - \begin{align*} - \vc^*(\Phi) &= \text{degree of polynomial growth of $\pi^*_\Phi(n)$} = \limsup_{n \to \infty}\frac{\log \pi^*_\Phi(n)}{\log n} - \end{align*} -\end{Lemma} - -One can check that the shatter function and hence VC-dimension and vc-density of a formula are elementary notions, -so they only depend on the first-order theory of the structure. - -NIP theories are a natural context for studying vc-density. -In fact we can take the following as the definition of NIP: -\begin{Definition} - Define $\phi$ to be NIP if it has finite VC-dimension. -\end{Definition} - -% \cite{Aschenbrenner_reference_8} shows that in a general combinatorial context, -In a general combinatorial context, -vc-density can be any real number in $0 \cup [1, \infty)$. -Less is known if we restrict our attention to NIP theories. -Proposition 4.6 in \cite{vc_density} gives examples of formulas that have non-integer rational vc-density in an NIP theory, -however it is open whether one can get an irrational vc-density in this context. - -In general, instead of working with a theory formula by formula, we can look for a uniform bound for all formulas: -\begin{Definition} - For a given NIP structure $M$, define the \defn{vc-function} - \begin{align*} - \vc^M(n) &= \sup \{\vc^*(\phi(x, y)) \mid \phi \in \LL(M), |x| = n\} \\ - &= \sup \{\vc(\phi(x, y)) \mid \phi \in \LL(M), |y| = n\} - \end{align*} -\end{Definition} - -As before this definition is elementary, so it only depends on the theory of $M$. -We omit the superscript $M$ if it is understood from the context. -One can easily check the following bounds: -\begin{Lemma} [Lemma 3.22 in \cite{vc_density}] - \begin{align*} - \vc(1) &\geq 1 \\ - \vc(n) &\geq n\vc(1) - \end{align*} -\end{Lemma} - -However, it is not known whether the second inequality can be strict or even whether $\vc(1) < \infty$ implies $\vc(n) < \infty$. +\input{../vc_intro.tex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Graph Combinatorics} @@ -712,7 +564,7 @@ \section{Upper bound} \begin{thebibliography}{9} -\bibitem{vc_density} +\bibitem{density} M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko, \textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, Trans. Amer. Math. Soc. 368 (2016), 5889-5949 diff --git a/research/10 QP reduct/#QP_reduct.tex# b/research/10 QP reduct/#QP_reduct.tex# new file mode 100644 index 00000000..5a56d5ef --- /dev/null +++ b/research/10 QP reduct/#QP_reduct.tex# @@ -0,0 +1,1007 @@ +\documentclass{amsart} + +\usepackage{../AMC_style} +\usepackage{../Research} +\usepackage{../Thm} + +\usepackage{enumitem} + +\usepackage{mathrsfs} +\usepackage{pgfpages} +\usepackage{setspace} + +\doublespacing +%\usepackage[margin=.75in]{geometry} +%\pgfpagesuselayout{2 on 1} + +\renewcommand{\AA}{\mathscr A} +\newcommand{\BB}{\mathscr B} +\newcommand{\DD}{\mathscr D} +\newcommand{\II}{\mathscr I} +\newcommand{\MM}{\mathscr M} + +\newcommand{\A}{\mathcal A} +\newcommand{\B}{\mathcal B} +\renewcommand{\C}{\mathcal C} +\newcommand{\D}{\mathcal D} +\newcommand{\F}{\mathcal F} +\newcommand{\G}{\mathcal G} +\renewcommand{\H}{\mathcal H} +\renewcommand{\LL}{\mathcal L} +\newcommand{\LLA}{\mathcal L_{aff}} +\newcommand{\LLM}{\mathcal L_{Mac}} +\newcommand{\M}{\mathcal M} + +\newcommand{\U}{\mathcal U} + + +\newcommand{\curly}[1]{\left\{#1\right\}} +\newcommand{\paren}[1]{\left(#1\right)} +\newcommand{\abs}[1]{\left|#1\right|} + +\providecommand{\floor}[1]{\left \lfloor #1 \right \rfloor } + +\DeclareMathOperator{\Sg}{Sg} +\DeclareMathOperator{\It}{Tp} +\DeclareMathOperator{\Sub}{Sub} +\DeclareMathOperator{\Ct}{Ct} +\DeclareMathOperator{\vecspan}{span} +\DeclareMathOperator{\val}{val} +\DeclareMathOperator{\vval}{val} +\DeclareMathOperator{\tval}{T-val} +\DeclareMathOperator{\inti}{I} + +\newcommand{\defn}{\underline} +\newcommand{\interval}{\inti(t, \alpha_L, \alpha_U)} + + + +\title{vc-density in an additive reduct of the $P$-adic numbers} +\author{Anton Bobkov} +\email{bobkov@math.ucla.edu} + +\begin{document} + +\begin{abstract} + Aschenbrenner et. al. computed a bound $\vc(n) \leq 2n - 1$ for the vc-density function in the field of $p$-adic numbers, + but it is not known to be optimal. + In this paper we investigate a certain $P$-minimal additive reduct of the field of $p$-adic numbers and + use a cell decomposition result of Leenknegt to compute an optimal bound $\vc(n) = n$ for that structure. +\end{abstract} + + +\maketitle + +VC-density was studied in model theory in \cite{density} by Aschenbrenner, Dolich, Haskell, MacPherson, and Starchenko +as a natural notion of dimension for definable families of sets in NIP theories. +In a complete NIP theory $T$ we can define the vc-function + +\begin{align*} + \vc^T = \vc : \N \arr \R \cup \curly{\infty} +\end{align*} + +where $\vc(n)$ measures the worst-case complexity of families of definable sets in an $n$-fold cartesian power of the underlying set of a model of $T$ +(see \ref{vc_fn_def} below for a precise definition of $vc^T$). +The simplest possible behavior is $\vc(n) = n$ for all $n$, +satisfied, for example, if $T$ is o-minimal. +For $T = \Th(\Q_p)$, the paper \cite{density} computes an upper bound for this function to be $2n-1$, and it is not known whether this is optimal. +This same bound holds in any reduct of the field of $p$-adic numbers, but one may expect that the simplified structure of the reduct would allow a better bound. +In \cite{reduct}, Leenknegt provides a cell decomposition result for a certain $P$-minimal additive reduct of the field of $p$-adic numbers. +Using this result, in this paper we improve the bound for the vc-function, showing that in Leenknegt's structure $\vc(n) = n$. + +Section 1 defines vc-density and states some basic lemmas about it. +A more in depth exposition of vc-density can be found in \cite{density}. +Section 2 defines and states some basic facts about the theory of $p$-adic numbers. +Here we also introduce the reduct which we will be working with. +Section 3 sets up basic definitions and lemmas that will be needed for the proof. +We define trees and intervals and show how they help with vc-density calculations. +Section 4 concludes the proof. + +Throughout the paper, variables and tuples of elements will be simply denoted as $x, y, a, b, \ldots$. +We will occasionally write $\vec a$ instead of $a$ for a tuple in $\Q_p^n$ to emphasize it as an element of $\Q_p$-vector space $\Q_p^n$. +We denote the arity of a tuple $x$ of variables by $|x|$. +% First-order formulas will have parameter variables separated $\phi(x; y)$. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{VC-dimension and vc-density} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + Throughout this section we work with a collection $\F$ of subsets of a set $X$. + We call the pair $(X, \F)$ a \defn{set system}. + +\begin{Definition} + \begin{itemize} + \item Given a subset $A$ of $X$, we define the set system $(A, A \cap \F)$ + where $A \cap \F = \curly{A \cap F \mid F\in \F}$. + \item For $A \subset X$ we say that $\F$ \defn{shatters} $A$ if $A \cap \F = \PP(A)$ (the power set of $A$). + \end{itemize} +\end{Definition} + +\begin{Definition} + We say $(X, \F)$ has \defn{VC-dimension} $n$ if the largest subset of $X$ shattered by $\F$ is of size $n$. + If $\F$ shatters arbitrarily large subsets of $X$, we say that $(X, \F)$ has infinite VC-dimension. + We denote the VC-dimension of $(X, \F)$ by $\VC(X, \F)$. +\end{Definition} + +\begin{Note} + We may drop $X$ from the $\VC(X, \F)$, as the VC-dimension doesn't depend on the base set and is determined by $(\bigcup \F, \F)$. +\end{Note} +Set systems of finite VC-dimension tend to have good combinatorial properties, +and we consider set systems with infinite VC-dimension to be poorly behaved. + +Another natural combinatorial notion is that of a dual system: +\begin{Definition} + For $a \in X$ define $X_a = \curly{F \in \F \mid a \in F}$. + Let $\F^* = \curly{X_a \mid a \in X}$. + We call $(\F, \F^*)$ the \defn{dual system} of $(X, \F)$. + The VC-dimension of the dual system of $(X, \F)$ is referred to as the \defn{dual VC-dimension} of $(X, \F)$ and denoted by $\VC^*(\F)$. + (As before, this notion doesn't depend on $X$.) +\end{Definition} + +\begin{Lemma} + A set system $(X, \F)$ has finite VC-dimension if and only if its dual system has finite VC-dimension. + More precisely + \begin{align*} + \VC^*(\F) \leq 2^{1+\VC(\F)}. + \end{align*} +\end{Lemma} + +For a more refined notion of complexity of $(X, \F)$ we look at the traces of our family on finite sets: +\begin{Definition} + Define the \defn{shatter function} $\pi_\F \colon \N \arr \N$ and the \defn{dual shatter function} $\pi^*_\F \colon \N \arr \N$ of $\F$ by + \begin{align*} + \pi_\F(n) &= \max \curly{|A \cap \F| \mid A \subset X \text{ and } |A| = n} \\ + \pi^*_\F(n) &= \max \curly{\text{atoms($B$)} \mid B \subset \F, |B| = n} + \end{align*} + where atoms($B$) = number of atoms in the Boolean algebra of sets generated by $B$. + Note that the dual shatter function is precisely the shatter function of the dual system: $\pi^*_\F = \pi_{\F^*}$. +\end{Definition} + +A simple upper bound is $\pi_\F(n) \leq 2^n$ (same for the dual). +If the VC-dimension of $\F$ is infinite then clearly $\pi_\F(n) = 2^n$ for all $n$. Conversely we have the following remarkable fact: +\begin{Theorem} [Sauer-Shelah '72] + If the set system $(X, \F)$ has finite VC-dimension $d$ then $\pi_\F(n) \leq {n \choose \leq d}$ for all $n$, where + ${n \choose \leq d} = {n \choose d} + {n \choose d - 1} + \ldots + {n \choose 1}$. +\end{Theorem} + +Thus the systems with a finite VC-dimension are precisely the systems where the shatter function grows polynomially. +Define the vc-density of $\F$ to quantify the growth of the shatter function of $\F$: +\begin{Definition} + Define \defn{vc-density} and \defn{dual vc-density} of $\F$ as + \begin{align*} + \vc(\F) &= \limsup_{n \to \infty}\frac{\log \pi_\F(n)}{\log n} \in \R^{\geq 0} \cup \curly{+\infty},\\ + \vc^*(\F) &= \limsup_{n \to \infty}\frac{\log \pi^*_\F(n)}{\log n}\in \R^{\geq 0} \cup \curly{+\infty}. + \end{align*} +\end{Definition} + +Generally speaking a shatter function that is bounded by a polynomial doesn't itself have to be a polynomial. +Proposition 4.12 in \cite{density} gives an example of a shatter function that grows like $n \log n$ (so it has vc-density $1$). + +So far the notions that we have defined are purely combinatorial. +We now adapt VC-dimension and vc-density to the model theoretic context. + +\begin{Definition} + Work in a first-order structure $M$. + Fix a finite collection of formulas $\Phi(x, y)$. + + \begin{itemize} + \item For $\phi(x, y) \in \LL(M)$ and $b \in M^{|y|}$ let + \begin{align*} + \phi(M^{|x|}, b) = \{a \in M^{|x|} \mid \phi(a, b)\} \subseteq M^{|x|}. + \end{align*} + \item Let $\Phi(M^{|x|}, M^{|y|})= \{\phi(M^{|x|}, b) \mid \phi_i \in \Phi, b \in M^{|y|}\} \subseteq \PP(M^{|x|})$. + \item Let $\F_\Phi = \Phi(M^{|x|}, M^{|y|})$, giving rise to a set system $(M^{|x|}, \F_\Phi)$. + \item Define the \defn{VC-dimension} of $\Phi$, $\VC(\Phi)$ to be the VC-dimension of $(M^{|x|}, \F_\Phi)$, similarly for the dual. + \item Define the \defn{vc-density} of $\Phi$, $\vc(\Phi)$ to be the vc-density of $(M^{|x|}, \F_\Phi)$, similarly for the dual. + \end{itemize} + + We will also refer to the vc-density and VC-dimension of a single formula $\phi$ + viewing it as a one element collection $\Phi = \curly{\phi}$. +\end{Definition} + +Counting atoms of a Boolean algebra in a model theoretic setting corresponds to counting types, +so it is instructive to rewrite the shatter function in terms of types. + +\begin{Definition} + \begin{align*} + \pi^*_\Phi(n) &= \max \curly{\text{number of $\Phi$-types over $B$} \mid B \subset M, |B| = n} + \end{align*} + Here a $\Phi$-type over $B$ is a maximal consistent collection of functions of the form $\phi(x, b)$ or $\neg\phi(x, b)$ + where $\phi \in \Phi$ and $b \in B$. +\end{Definition} + +\begin{Lemma} \label{count_types} + \begin{align*} + \vc^*(\Phi) &= \text{degree of polynomial growth of $\pi^*_\Phi(n)$} = \limsup_{n \to \infty}\frac{\log \pi^*_\Phi(n)}{\log n} + \end{align*} +\end{Lemma} + +\begin{proof} + \begin{align*} + &\pi^*_{\F_\Phi}\paren{n} \leq \pi^*_\Phi(n) \leq \pi^*_{\F_\Phi}\paren{|\Phi|n} \\ + &\vc^*(\Phi) \leq \limsup_{n \to \infty}\frac{\log \pi^*_\Phi(n)}{\log n} \leq \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{|\Phi|n}}{\log n} = \\ + & = \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{|\Phi|n}}{\log |\Phi|n} \frac{\log |\Phi|n}{\log n} = + \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{|\Phi|n}}{\log |\Phi|n} \leq \\ + &\leq \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{n}}{\log n} = \vc^*(\Phi) + \end{align*} +\end{proof} + +One can check that the shatter function and hence VC-dimension and vc-density of a formula are elementary notions, +so they only depend on the first-order theory of the structure. + +NIP theories are a natural context for studying vc-density. +In fact we can take the following as the definition of NIP: +\begin{Definition} + Define $\phi$ to be NIP if it has finite VC-dimension. + A theory $T$ is NIP if all the formulas are NIP. +\end{Definition} + +% \cite{Aschenbrenner_reference_8} shows that in a +% \item general combinatorial context, +In a general combinatorial context for arbitrary set systems, +vc-density can be any real number in $0 \cup [1, \infty)$. +Less is known if we restrict our attention to NIP theories. +Proposition 4.6 in \cite{density} gives examples of formulas that have non-integer rational vc-density in an NIP theory, +however it is open whether one can get an irrational vc-density in this model-theoretic setting. + +Instead of working with a theory formula by formula, we can look for a uniform bound for all formulas: +\begin{Definition} \label{vc_fn_def} + For a given NIP structure $M$, define the \defn{vc-function} + \begin{align*} + \vc^M(n) &= \sup \{\vc^*(\phi(x, y)) \mid \phi \in \LL(M), |x| = n\} \\ + &= \sup \{\vc(\phi(x, y)) \mid \phi \in \LL(M), |y| = n\} + \end{align*} +\end{Definition} + +As before this definition is elementary, so it only depends on the theory of $M$. +We omit the superscript $M$ if it is understood from the context. +One can easily check the following bounds: +\begin{Lemma} [Lemma 3.22 in \cite{density}] We have $\vc(1) \geq 1$ and $\vc(n) \geq n\vc(1)$. + +\end{Lemma} + +However, it is not known whether the second inequality can be strict or even whether $\vc(1) < \infty$ implies $\vc(n) < \infty$. + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{$P$-adic numbers} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +The field $\Q_p$ of $p$-adic numbers is often studied in the language of Macintyre + \begin{align*} + \LLM = \curly{0, 1, +, -, \cdot, |, \{P_n\}_{n \in \N}} + \end{align*} +which is a language $\curly{0, 1, +, -, \cdot}$ of fields together with unary predicates $P_n$ interpreted in $\Q_p$ so as to satisfy + +\begin{align*} + P_n x \leftrightarrow \exists y \; y^n = x +\end{align*} + +and a divisibility relation where $a|b$ holds in $\Q_p$ when $\vval a \leq \vval b$. + +Note that $P_n\backslash \curly{0}$ is a multiplicative subgroup of $\Q_p$ with finitely many cosets. + +\begin{Theorem} [Macintyre '76] + The $\LLM$-structure $\Q_p$ has quantifier elimination. +\end{Theorem} + +There is also a cell decomposition result: +\begin{Definition} + Define \defn{$k$-cell} recursively as follows. + $0$-cell is a singleton subset of $\Q_p$. + A $(k+1)$-cell is a subset of $\Q_p^{k+1}$ of the following form: + \begin{align*} + \curly{(x, t) \in D \times \Q_p \mid \vval a_1(x) \ \square_1 \vval (t - c(x)) \ \square_2 \vval a_2(x), t - c(x) \in \lambda P_n} + \end{align*} + where $D$ is a $k$-cell, + $a_1(x), a_2(x), c(x)$ are definable functions $D \arr \Q_p$, + $\square_i$ is $<, \leq$ or no condition, and + $\lambda \in \Q_p$. +\end{Definition} + +\begin{Theorem} [Denef '84] + Any definable subset of $Q_p^n$ defined by an $\LLM$-formula decomposes into a finite disjoint union of $n$-cells. +\end{Theorem} + +In \cite{density}, Aschenbrenner, Dolich, Haskell, Macpherson, and Starchenko show that this structure satisfies $\vc(n) \leq 2n - 1$, +however it is not known whether this bound is optimal. + +In \cite{reduct}, Leenknegt analyzes the reduct of $\Q_p$ to the language +\begin{align*} + \LL_{aff} = \curly{0, 1, +, -, \curly{\bar c}_{c \in \Q_p}, |, \curly{Q_{m,n}}_{m,n\in \N}} +\end{align*} +where $\bar c$ denotes a scalar multiplication by $c$, +$a | b$ as above stands for $\vval a \leq \vval b$, +and $Q_{m,n}$ is a unary predicate interpreted as +\begin{align*} + Q_{m,n} = \bigcup_{k \in \Z} p^{km} (1 + p^n\Z_p). +\end{align*} +Note that $Q_{m,n} \backslash \curly{0}$ is a subgroup of the multiplicative group of $\Q_p$ with finitely many cosets. +One can check that these extra relation symbols are definable in the $\LLM$-structure $\Q_p$. +The paper \cite{reduct} provides a cell decomposition result with the following cells: + +\begin{Definition} \label{cell} + A $0$-cell is a singleton subset of $\Q_p$. + A $(k+1)$-cell is a subset of $\Q_p^{k+1}$ of the following form: + \begin{align*} + \curly{(x, t) \in D \times \Q_p \mid \vval a_1(x) \ \square_1 \vval (t - c(x)) \ \square_2 \vval a_2(x), t - c(x) \in \lambda Q_{m,n} } + \end{align*} + where $D$ is a $k$-cell, called the \defn{base} of the cell, + $a_1(x), a_2(x), c(x)$ are polynomials of degree $\leq 1$, called the \defn{defining polynomials} + $\square_1, \square_2$ is $<$ or no condition, and + $\lambda \in\Q_p$. + We call $\Q_{m,n}$ the \defn{defining predicate}. +\end{Definition} + +\begin{Theorem}[Leenknegt '12] + Any definable subset of $Q_p^n$ defined by an $\LL_{aff}$-formula decomposes into a finite disjoint union of $n$-cells. + %Any formula $\phi(x, t)$ in $(\Q_p, \LL_{aff})$ with $|x| = n$ and $|t| = 1$ decomposes into a union of $(k+1)$-cells. +\end{Theorem} + +Moreover, \cite{reduct} shows that $(\Q_p, \LL_{aff})$ is a $P$-minimal reduct, +that is, the one-dimensional definable sets of $(\Q_p, \LL_{aff})$ coincide with the one-dimensional definable sets in the full structure $(\Q_p, \LLM)$. + +The main result of this paper is the computation of the $\vc$-function for this structure: +\begin{Theorem} \label{main_theorem} + $(\Q_p, \LL_{aff})$ has $\vc(n) = n$. +\end{Theorem} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Key Lemmas and Definitions} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +To show that $\vc(n) = n$ it suffices to bound $\vc^*(\phi) \leq |x|$ for every $\LL_{aff}$-formula $\phi(x; y)$. +Fix such a formula $\phi(x; y)$. +Instead of working with it directly, we simplify it using quantifier elimination. +The required quantifier elimination result can be easily obtained from cell decomposition: +\begin{Lemma} \label {quantifier_elimination} + Any formula $\phi(x; y)$ in $(\Q_p, \LL_{aff})$ can be written as a boolean combination of formulas from the following collection + \begin{align*} + \Phi(x; y) = &\curly{\vval (p_i(x) - c_i(y)) < \vval (p_j(x) - c_j(y))}_{i, j \in I} \cup \\ + &\curly{p_i(x) - c_i(y) \in \lambda_k Q_{m,n}}_{i \in I , k \in K} + \end{align*} + where $I, K$ are finite index sets, + each $p_i$ is a degree $\leq 1$ polynomial in $x$ without a constant term, + each $c_i$ is a degree $\leq 1$ polynomial in $y$, and + $\lambda_k \in \Q_p$. +\end{Lemma} + +\begin{proof} + Let $l = |x| + |y|$. + Partition the subset of $\Q_p^l$ defined by $\phi$ to obtain $\DD^l$, a collection of $l$-cells. + Let $\DD^{l-1}$ be the collection of the bases of the cells in $\DD^l$. + Similarly, construct by induction $\DD^i$ for each $0 \leq j < l$, + where $\DD^j$ is the collection of $j$-cells which are the bases of cells in $\DD^{j+1}$. + % Let $\DD = \bigcup \DD_j$. + \begin{align*} + &m = \prod \curly{m' \mid Q_{m',n'} \text{ is the defining predicate of a cell in $\DD^j$ for $0 \leq j \leq l$} } \\ + &n = \max \curly{n' \mid Q_{m',n'} \text{ is the defining predicate of a cell in $\DD^j$ for $0 \leq j \leq l$} } + \end{align*} + % Choose $m,n$ large enough to cover all $m', n'$ for $Q_{m',n'}$ that show up in the cells of $\DD$. + This way if $a, a'$ are in the same coset of $Q_{m',n'}$ then they are in the same coset of $Q_{m,n}$. + Choose $\curly{\lambda_k}_{k \in K}$ to go over all the cosets of $Q_{m,n}$. + Let $q_i(x, y)$ enumerate all of the defining polynomials $a_1(x), a_2(x), t - c(x)$ that show up in the cells of $\DD^j$ for any $j$. + All if those are all polynomials of degree $\leq 1$ in variables $x, y$. + We can split each of them as $q_i(x,y) = p_i(x) - c_i(y)$ where the constant term goes into $c_i$. + This gives us the appropriate finite collection of formulas $\Phi$. + From the cell decomposition it is easy to see that when $a, a'$ have the same $\Phi$-type, + then they have the same $\phi$-type. + Thus $\phi$ can be written as a boolean combination of formulas from $\Phi$. +\end{proof} + +\begin{Lemma} + Let $\Phi(x; y)$ be a finite collection of formulas. + If $\phi$ can be written as a boolean combination of formulas from $\Phi$ then + \begin{align*} + \vc^* (\Phi) \leq r \implies \vc^* (\phi) \leq r \; \text{ for all } r \in \R + \end{align*} +\end{Lemma} +\begin{proof} + If $a,a'$ have the same $\Phi$-type over $B$, then they have the same $\phi$-type over $B$, where $B$ is some parameter set. + Therefore the number of $\phi$-types is bounded by the number of $\Phi$-types. + The bound follows from Lemma \ref{count_types}. +\end{proof} + +For the remainder of the paper fix $\Phi(x; y)$ to be the collection of formulas defined by Lemma \ref{quantifier_elimination}. +By the previous lemma, to show that $\vc^*(\phi) \leq |x|$, it suffices to bound $\vc^*(\Phi) \leq |x|$. +More precisely, it is sufficient to show that if there is a parameter set $B$ of size $N$ +then the number of $\Phi$-types over $B$ is $O(N^{|x|})$. +Fix such a parameter set $B$ and work with it from now on. +We will compute a bound for the number of $\Phi$-types over $B$. + +Consider a set $T = T(\Phi, B) = \curly{c_i(b) \mid b \in B, i \in I} \subset \Q_p$. +In this definition $B$ is the parameter set that we have fixed +and $c_i(b)$ come from the collection of formulas $\Phi$ from the quantifier elimination above. +View $T$ as a tree as follows: +\begin{Definition} \ + \begin{itemize} + \item For $c \in \Q_p, \alpha \in \Z$ define a \defn{ball} + \begin{align*} + B(c, \alpha) = \curly{c' \in \Q_p \mid \vval \paren{c' - c} > \alpha}. + \end{align*} + We also let $B(c, -\infty) = \Q_p$ and $B(c, +\infty) = \emptyset$. + \item Define a collection of balls $\BB = \curly{B(t_1, \vval(t_1 - t_2))}_{t_1, t_2 \in T}$. + Note that $\BB$ is a (directed) boolean algebra of sets in $\Q_p$. + We refer to the atoms in that algebra as \defn{intervals}. + Note that the intervals partition $\Q_p$ so any element $a \in \Q_p$ belongs to a unique interval. + \item Let's introduce some notation for the intervals. + For $t \in T$ and $\alpha_L, \alpha_U \in \Z \cup \curly{-\infty, +\infty}$ define + \begin{align*} + \interval = B(t, \alpha_L) \backslash \bigcup \curly{B(t', \alpha_U) \mid t' \in T, \vval(t' - t) \geq \alpha_U} + \end{align*} + (this is sometimes referred to as the swiss cheese construction). + One can check that every interval is of the form $\interval$ for some values of $t, \alpha_L, \alpha_U$. + Quantities $\alpha_L, \alpha_U$ are uniquely determined by the interval, + while $t$ might not be. + \item Intervals are a natural construction for trees, however we will require a more refined notion to make Lemma \ref{main_lemma} below work. + Define a larger collection of balls + \begin{align*} + \BB' = \BB \cup \curly{B(c_i(b), \vval(c_j(b) - c_k(b)))}_{i,j,k \in I, b \in B}. + \end{align*} + Similar to the previous definition, we define a \defn{subinterval} to be an atom of the boolean algebra generated by $\BB'$. + Subintervals refine intervals. + Moreover, as before, each subinterval can be written as $\interval$ for some values of $t, \alpha_L, \alpha_U$. + As before, $\alpha_L, \alpha_U$ are uniquely determined by the subinterval, + while $t$ might not be. + \end{itemize} +\end{Definition} + +Subintervals are fine enough to make Lemma \ref{main_lemma} below work while coarse enough to be $O(N)$ small: +\begin{Lemma} \label{interval_count}\ + \begin{itemize} + \item + There are at most $2|T| = 2 N |I| = O(N)$ different intervals. + \item + There are at most $2|T| + |B| \cdot |I|^3 = O(N)$ different subintervals. + \end{itemize} +\end{Lemma} + +\begin{proof} + Each new element in the tree $T$ adds at most two intervals to the total count, + so by induction there can be at most $2|T|$ many intervals. + Each new ball in $\BB' \backslash \BB$ adds at most one subinterval to the total count, + so by induction there are at most $|\BB' \backslash \BB|$ more subintervals than there are intervals. +\end{proof} + + +\begin{Definition} + Suppose $a \in \Q_p$ lies in an interval $\interval$. + Define the \defn{T-valuation} of $a$ to be $\tval(a) = \vval(a - t)$. +\end{Definition} + +This a natural notion having the following properties: +\begin{Lemma} \label{tval} \ + \begin{enumerate}[label=(\alph*)] + \item $\tval(a)$ is well-defined, independent of choice of $t$ to represent the interval. + \item If $a \in \Q_p$ lies in a subinterval $\interval$, + then $\tval(a) = \vval(a - t)$. + \item If $a \in \Q_p$ lies in a (sub)interval $\interval$ + then $\alpha_L < \tval(a) \leq \alpha_U$. + \item For any $a \in \Q_p$ lying in a (sub)interval $\interval$ and $t' \in T$ + \begin{itemize} + \item If $\vval(t - t') \geq \alpha_U$, then $\vval(a - t') = \tval(a)$. + \item If $\vval(t - t') \leq \alpha_L$, then $\vval(a - t') = \vval(t - t') \paren{\leq \alpha_L < \tval(a)}$. + \end{itemize} + \end{enumerate} +\end{Lemma} + + +\begin{proof} + (a)-(c) are clear. + For (d) fix $t' \in T$ and suppose $a \in \Q_p$ lies in a subinterval $\inti(t, \alpha_L', \alpha_U')$. + This subinterval lies inside of an interval $\interval$ for some choice of $\alpha_L, \alpha_U$ and + by the definition of intervals (or more specifically $\BB$) + \begin{align*} + \vval(t - t') \geq \alpha_U &\iff \vval(t - t') \geq \alpha_U'\\ + \vval(t - t') \geq \alpha_L &\iff \vval(t - t') \geq \alpha_L'. + \end{align*} + Therefore without loss of generality we may assume that $a \in \Q_p$ lies in an interval $\interval$. + By (c) and the definition of intervals one of the three following cases has to hold. + + Case 1: $\vval(t - t') \geq \alpha_U$ and $\tval(a) < \alpha_U$. Then + \begin{align*} + \vval(t - t') \geq \alpha_U > \tval(a) = \vval(a - t), + \end{align*} + thus $\vval(a - t') = \vval(a - t) = \tval(a)$ as needed. + + Case 2: $\vval(t - t') \geq \alpha_U$ and $\tval(a) = \alpha_U$. Then + \begin{align*} + \tval(a) = \vval(a - t) = \vval(t - t') \geq \alpha_U, + \end{align*} + thus $\vval(a - t') \geq \alpha_U$. + The interval $\interval$ is disjoint from the ball $B(t', \alpha_U)$, + so $a \notin B(t', \alpha_U)$, that is, $\val(a - t') \leq \alpha_U$. + Combining this with the previous inequality we get that $\val(a - t') = \alpha_U = \tval(a)$ as needed. + + Case 3: $\vval(t - t') \leq \alpha_L$. Then + \begin{align*} + \vval(t - t') \leq \alpha_L < \tval(a) = \vval(a - t), + \end{align*} + thus $\vval(a - t') = \vval(t - t')$ as needed. +\end{proof} + + + + +\begin{Definition} + Suppose $a \in \Q_p$ lies in a subinterval $\interval$. + We say that $a$ is \defn{far from the boundary} (of $\interval$) if + \begin{align*} + \alpha_L + n \leq \tval(a) \leq \alpha_U - n. + \end{align*} + Here $n$ is from the Lemma \ref{quantifier_elimination}. + Otherwise we say that it is \defn{close to the boundary}. +\end{Definition} + +\begin{Definition} + Suppose $a_1, a_2 \in \Q_p$ lie in the same subinterval $\interval$. + We say $a_1, a_2$ have the same \defn{subinterval type} if one of the following holds: + \begin{itemize} + \item Both $a_1, a_2$ are far from the boundary and $a_1 - t, a_2 - t$ are in the same $Q_{m,n}$-coset. + ($Q_{m,n}$ is from the Lemma \ref{quantifier_elimination}.) + \item Both $a_1, a_2$ are close to the boundary and + \begin{align*} + \tval(a_1) = \tval(a_2) \leq \vval(a_1 - a_2) - n. + \end{align*} + \end{itemize} +\end{Definition} + + +\begin{Definition} + For $c \in \Q_p$ and $\alpha, \beta \in \Z, \alpha < \beta$ define $c \midr [\alpha, \beta) \in \paren{\Z/p\Z}^{\beta - \alpha}$ + to be the record of the coefficients of $c$ for the valuations between $[\alpha, \beta)$. + More precisely write $c$ in its power series form + \begin{align*} + c = \sum_{\gamma \in \Z} c_\gamma p^\gamma \text{ with } c_\gamma \in \curly{0,1, \ldots, p-1} + \end{align*} + Then $c \midr [\alpha, \beta)$ is just $(c_\alpha, c_{\alpha+1}, \ldots c_{\beta - 1})$. +\end{Definition} + +The following lemma is an adaptation of Lemma 7.4 in \cite{density}. +\begin{Lemma} \label{distance} + Fix $m,n \in \N$. + For any $x,y,c \in \Q_p$, if + \begin{align*} + \val (x - c) = \val (y - c) \leq \val (x - y) - n, + \end{align*} + then $x - c, y - c$ are in the same coset of $Q_{m,n}$. +\end{Lemma} +\begin{proof} + Call $a,b \in \Q_p$ similar if $\val a = \val b$ and + \begin{align*} + a \midr [\val a, \val a + n) = b \midr [\val b, \val b + n) + \end{align*} + If $a,b$ are similar then + \begin{align*} + a \in Q_{m,n} \iff b \in Q_{m,n} + \end{align*} + Moreover for any $\lambda \in \Q_p^\times$, if $a,b$ are similar then so are $\lambda a, \lambda b$. + Thus if $a,b$ are similar, then they belong to the same coset of $Q_{m,n}$. + Conditions of the lemma force $x - c, y - c$ to be similar, thus belonging to the same coset. +\end{proof} + + +\begin{Lemma} \label{interval_type_count} + For each subinterval there are at most $K = K(Q_{m,n})$ many subinterval types + (with $K$ not depending on $B$ or on the subinterval). +\end{Lemma} + +\begin{proof} + Let $a, a' \in \Q_p$ lie in the same subinterval $\interval$. + + Suppose $a, a'$ are far from the boundary. + Then they have the same subinterval type if $a - t, a' - t$ are in the same $Q_{m,n}$-coset. + The number of such subinterval types is bounded by the number of $Q_{m,n}$-cosets. + + Suppose $a, a'$ are close to the boundary and + \begin{align*} + &\tval(a) - \alpha_L = \tval(a') - \alpha_L < n \text { and}\\ + &a \midr [\tval(a), \tval(a) + n) = a' \midr [\tval(a'), \tval(a') + n) + \end{align*} + Then $a, a'$ have the same subinterval type. + Such a subinterval type is thus determined by $\tval(a) - \alpha_L$ and $a \midr [\tval(a), \tval(a) + n)$, + therefore there are at most $n p^n$ many such types. + + A similar argument works for $a$ with $\alpha_U - \tval(a) \leq n$. + + Adding those up we get that there are at most + \begin{align*} + K = \text{(number of $Q_{m,n}$ cosets)} + 2 n p^n + \end{align*} + many subinterval types. +\end{proof} + +The following critical lemma relates tree notions to $\Phi$-types. +\begin{Lemma} \label{main_lemma} + Suppose $d, d' \in \Q_p^{|x|}$ satisfy the follwing three conditions: + \begin{itemize} + \item For all $i \in I$ $p_i(d)$ and $p_i(d')$ are in the same subinterval. + \item For all $i \in I$ $p_i(d)$ and $p_i(d')$ have the same subinterval type. + \item For all $i,j \in I$, $\tval(p_i(d)) > \tval(p_j(d))$ iff $\tval(p_i(d')) > \tval(p_j(d'))$. + \end{itemize} + Then $d, d'$ have the same $\Phi$-type over $B$. +\end{Lemma} +\begin{proof} + There are two kinds of formulas in $\Phi$ + (see Lemma \ref{quantifier_elimination}). + First we show that $d, d'$ agree on formulas of the form $p_i(x) - c_i(y) \in \lambda_k Q_{m,n}$. + It is enough to show that for every $i \in I, b \in B$, $p_i(d) - c_i(b), p_i(d') - c_i(b)$ are in the same $Q_{m,n}$-coset. + Fix such $i, b$. + For brevity let $a = p_i(d), a' = p_i(d')$ and $Q = Q_{m,n}$. + We want to show that $a - c_i(b), a' - c_i(b)$ are in the same $Q$-coset. + + Suppose $a, a'$ are close to the boundary. + Then $\tval(a) = \tval(a') \leq \val(a - a') - n$. + Using Lemma \ref{tval}d, we have + \begin{align*} + \val(a - c_i(b)) = \val(a' - c_i(b)) \leq \tval(a) \leq \val(a - a') - n. + \end{align*} + Lemma \ref{distance} shows that $a - c_i(b), a' - c_i(b)$ are in the same $Q$-coset. + + Now, suppose both $a, a'$ are far from the boundary. + Label their interval as $\interval$. + Then we have + \begin{align*} + \alpha_L + n \leq &\val (a - t) \leq \alpha_U - n \\ + \alpha_L + n \leq &\val (a' - t) \leq \alpha_U - n + \end{align*} + (as being far from the subinterval's boundary also makes $a,a'$ far from interval's boundary). + We have either $\val(t - c_i(b)) \geq \alpha_U$ or $\val(t - c_i(b)) \leq \alpha_L$ (as otherwise it would contradict the definition of intervals, or more specifically $\BB$). + + Suppose it is the first case $\val(t - c_i(b)) \geq \alpha_U$. + Then using Lemma \ref{tval}d + \begin{align*} + \val(a - c_i(b)) = \val(a - t) \leq \alpha_U - n \leq \val(t - c_i(b)) - n. + \end{align*} + So by Lemma \ref{distance} $a - c_i(b), a - t$ are in the same $Q$-coset. + By an analogous argument, $a' - c_i(b), a' - t$ are in the same $Q$-coset. + As $a, a'$ have the same subinterval type, $a - t, a' - t$ are in the same $Q$-coset. + Thus by transitivity we get that $a - c_i(b), a' - c_i(b)$ are in the same $Q$-coset. + + For the second case, suppose $\val(t - c_i(b)) \leq \alpha_L$. + Then using Lemma \ref{tval}d + \begin{align*} + \val(a - c_i(b)) = \val(t - c_i(b)) \leq \alpha_L \leq \val(a - t) - n, + \end{align*} + so by Lemma \ref{distance}, $a - c_i(b), t - c_i(b)$ are in the same $Q$-coset. + Similarly $a' - c_i(b), t - c_i(b)$ are in the same $Q$-coset. + Thus by transitivity we get that $a - c_i(b), a' - c_i(b)$ are in the same $Q$-coset. + + Next, we need to show that $d, d'$ agree on formulas of the form + $\vval (p_i(x) - c_i(y)) < \vval (p_j(x) - c_j(y))$ + (again, referring to the presentation in Lemma \ref{quantifier_elimination}). + Fix $i,j \in I, b \in B$. + We would like to show the following equivalence: + + \begin{equation} \label {eq:order_equation} + \vval (p_i(d) - c_i(b)) < \vval (p_j(d) - c_j(b)) \iff \vval (p_i(d') - c_i(b)) < \vval (p_j(d') - c_j(b)) + \end{equation} + + Suppose $p_i(d), p_i(d')$ are in the subinterval $\inti(t_i, \alpha_i, \beta_i)$ and + $p_j(d), p_j(d')$ are in the subinterval $\inti(t_j, \alpha_j, \beta_j)$. + Lemma \ref{tval}d yields the following four cases. + + Case 1: + \begin{align*} + &\vval (p_i(d) - c_i(b)) = \vval (p_i(d') - c_i(b)) = \vval(t_i - c_i(b)) \\ + &\vval (p_j(d) - c_j(b)) = \vval (p_j(d') - c_j(b)) = \vval(t_j - c_j(b)) + \end{align*} + Then it is clear that the equivalence \eqref{eq:order_equation} holds. + + Case 2: + \begin{align*} + &\vval (p_i(d) - c_i(b)) = \tval(p_i(d)) \text{ and } \vval (p_i(d') - c_i(b)) = \tval(p_i(d')) \\ + &\vval (p_j(d) - c_j(b)) = \tval(p_j(d)) \text{ and } \vval (p_j(d') - c_j(b)) = \tval(p_j(d')) + \end{align*} + Then the equivalence \eqref{eq:order_equation} holds by the third hypothesis of the lemma (that order of T-valuations is preserved). + + Case 3: + \begin{align*} + &\vval (p_i(d) - c_i(b)) = \vval (p_i(d') - c_i(b)) = \vval(t_i - c_i(b)) \\ + &\vval (p_j(d) - c_j(b)) = \tval(p_j(d)) \text{ and } \vval (p_j(d') - c_j(b)) = \tval(p_j(d')) + \end{align*} + If $p_j(d), p_j(d')$ are close to the boundary, + then $\tval(p_j(d)) = \tval(p_j(d'))$ and the equivalence \eqref{eq:order_equation} clearly holds. + Suppose then that $p_j(d), p_j(d')$ are far from the boundary. + \begin{align*} + \alpha_j + n \leq &\tval(p_j(d)), \tval(p_j(d')) \leq \beta_j - n \\ + \alpha_j < &\tval(p_j(d)), \tval(p_j(d')) < \beta_j + \end{align*} + and $\vval(t_i - c_i(b))$ lies outside of the $(\alpha_j, \beta_j)$ + by the definition of subinterval (more specifically definition of $\BB'$). + Therefore \eqref{eq:order_equation} has to hold. + (Note that we always have $\tval(p_j(d)), \tval(p_j(d')) \in (\alpha_j, \beta_j]$ by Lemma \ref{tval}c, so + we only need the far from the boundary condition to avoid the edge case of equality to $\beta_j$.) + + Case 4: + \begin{align*} + &\vval (p_i(d) - c_i(b)) = \tval(p_i(d)) \text{ and } \vval (p_i(d') - c_i(b)) = \tval(p_i(d')) \\ + &\vval (p_j(d) - c_j(b)) = \vval (p_j(d') - c_j(b)) = \vval(t_j - c_j(b)) + \end{align*} + Similar to case 3 (switching $i,j$). +\end{proof} + + + +\begin{Note} + The previous lemma gives us an upper bound on the number of types - there are at most $|2I|!$ many choices for the order of $\tval$, + $O(N)$ many choices for the subinterval for each $p_i$, + and $K$ many choices for the subinterval type for each $p_i$ (where $K$ is as in Lemma \ref{interval_type_count}), + giving a total of $O(N^{|I|}) \cdot K^{|I|} \cdot |I|! = O(N^{|I|})$ many types. + This implies $\vc^*(\Phi) \leq |I|$. + The biggest contribution to this bound are the choices among the $O(N)$ many subintervals for each $p_i$ with $i \in I$. + Are all of those choices realized? + Intuitively there are $|x|$ many variables and $|I|$ many equations, + so once we choose a subinterval for $|x|$ many $p_i$'s, the subintervals for the rest should be determined. + This would give the required bound $\vc^*(\Phi) \leq |x|$. + The next section outlines this idea formally. +\end{Note} + + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{Main Proof} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +An alternative way to write $p_i(c)$ is as a scalar product $\vec p_i \cdot \vec c$, +where $\vec p_i$ and $\vec c$ are vectors in $\Q_p^{|x|}$ (as $p_i(x)$ is homogeneous linear). + +\begin{Lemma} + Suppose we have a finite collection of vectors $\curly{\vec p_j}_{j \in J}$ with each $\vec p_j \in \Q_p^{|x|}$. + Suppose $\vec p \in \Q_p^{|x|}$ satisfies $\vec p \in \vecspan \curly{\vec p_j}_{j \in J}$, + and we have $\vec c \in \Q_p^{|x|}, \alpha \in \Z$ with $\val(\vec p_j \cdot \vec c) > \alpha \text{ for all } j \in J$. + Then $\val(\vec p \cdot \vec c) > \alpha - \gamma$ for some $\gamma \in \N$. + Moreover $\gamma$ can be chosen independently from $\vec c, \alpha$ depending only on $\curly{\vec p_j}_{j \in J}$. +\end{Lemma} + +\begin{proof} + For some $c_j \in \Q_p$ for $j \in J$ we have $\vec p = \sum_{j \in J} c_j \vec p_j$, + hence $\vec p \cdot \vec c = \sum_{j \in J} c_j \vec p_j \cdot \vec c$. + Thus + \begin{align*} + \val \paren{c_j \vec p_j \cdot \vec c} = \val \paren{c_j} + \val \paren{\vec p_j \cdot \vec c} > \val \paren{c_j} + \alpha. + \end{align*} + % Pick $\gamma = -\max \val \paren{c_i}$ or $0$ if all those values are positive. + Let $\gamma = \max(0, -\max_{j \in J} \val \paren{c_j})$. + % Let $\gamma = -\min(0, \max_{j \in J} \val \paren{c_j})$. + % Let $\gamma = \max(0, \min -\val \paren{c_j})$. + Then we have + \begin{align*} + &\val(\vec p \cdot \vec c) = + \val \paren{\sum_{j \in J} c_j \vec p_j \cdot \vec c} \geq \\ + \geq &\min_{j \in J} \val(\sum_{j \in J} c_j \vec p_j \cdot \vec c) > + \min_{j \in J} \val(c_j) + \alpha \geq + \alpha - \gamma + \end{align*} +\end{proof} + +\begin{Corollary} \label{gamma} + Suppose we have a finite collection of vectors $\curly{\vec p_i}_{i \in I}$ with each $\vec p_i \in \Q_p^{|x|}$. + Suppose $J \subset I$ and $i \in I$ satisfy $\vec p_i \in \vecspan \curly{\vec p_j}_{j \in J}$, + and we have $\vec c \in \Q_p^{|x|}, \alpha \in \Z$ with $\val(\vec p_j \cdot \vec c) > \alpha \text{ for all } j \in J$. + Then $\val(\vec p_i \cdot \vec c) > \alpha - \gamma$ + for some $\gamma \in \N$. + Moreover $\gamma$ can be chosen independently from $J, j, \vec c, \alpha$ depending only on $\curly{\vec p_i}_{i \in I}$. +\end{Corollary} +\begin{proof} + The previous lemma shows that we can pick such $\gamma$ for a given choice of $i, J$, but independent from $\alpha, \vec c$. + To get a choice independent from $i, J$, go over all such eligible choices + ($i$ ranges over $I$ and $J$ ranges over subsets of $I$), + pick $\gamma$ for each, and then take the maximum of those values. +\end{proof} + +Fix $\gamma$ according to Corollary \ref{gamma} corresponding to $\curly{\vec p_i}_{i \in I}$ given by our collection of formulas $\Phi$. +(The lemma above is a general result, but we only use it applied to the vectors given by $\Phi$.) + +\begin{Definition} + Suppose $a \in \Q_p$ lies in the subinterval $\interval$. + Define \defn{$T$-floor} of $a$ to be $F(a) = \alpha_L$. +\end{Definition} + +\begin{Definition} + Let $f: \Q_p^{|x|} \arr \Q_p^I$ with $f(c) = (p_i(c))_{i \in I}$. + Define the segment space $\Sg$ to be the image of $f$. +\end{Definition} + +Given a tuple $(a_i)_{i\in I}$ in the segment space, +look at the corresponding $T$-floors $\curly{F(a_i)}_{i\in I}$ and T-valuations $\curly{\tval(a_i)}_{i\in I}$. +Partition the segment space by the order types of $\{F(a_i)\}_{i\in I}$ and $\curly{\tval(a_i)}_{i\in I}$ (as subsets of $\Z$). + +Work in a fixed set $\Sg'$ of the partition. +After relabeling we may assume that +\begin{align*} + F(a_1) \geq F(a_2) \geq \ldots \text { for all $a_i \in \Sg'$} +\end{align*} + +Consider the (relabeled) sequence of vectors $\vec p_1, \vec p_2, \ldots, \vec p_I$. +There is a unique subset $J \subset I$ such that all vectors with indices in $J$ are linearly independent, +and all vectors with indices outside of $J$ are a linear combination of preceding vectors. +For any index $i \in I$ we call it \defn{independent} if $i \in J$ and we call it \defn{dependent} otherwise. + + +\begin{Definition} \ + \begin{itemize} + \item Denote $\Z/p\Z^\gamma$ as \defn{$\Ct$}. + Note that $|\Ct| = p^\gamma$. + \item Let \defn{$\It$} be the space of all subinterval types. + By Lemma \ref{interval_type_count} $|\It| \leq K$. + \item Let \defn{$\Sub$} be the space of all subintervals. + By Lemma \ref{interval_count} $|\Sub| \leq 3 |I|^2 \cdot N = O(N)$. + \end{itemize} +\end{Definition} + +\begin{Definition} + Now, we define the following function + \begin{align*} + g_{\Sg'}: \Sg' \arr \It^I \times \Sub^J \times \Ct^{I \backslash J} + \end{align*} + + Let $a = (a_i)_{i\in I} \in \Sg'$. + To define $g_{\Sg'}(a)$ we need to specify where it maps $a$ in each individual component of the product. + + For each $a_i$ record its subinterval type, giving the first component $\It^I$. + + For $a_j$ with $j \in J$, record the subinterval of $a_j$, giving the second component $\Sub^J$. + + For the third component $\Ct^{I \backslash J}$ do the following computation. + Pick $a_i$ with $i$ dependent. + Let $j$ be the largest independent index with $j < i$. + Record $a_i \midr [F(a_j) - \gamma, F(a_j))$. + + Combine $g_{\Sg'}$ for all the partitions to get a function + \begin{align*} + g: \Sg \arr \It^I \times \Sub^J \times \Ct^{I \backslash J}. + \end{align*} +\end{Definition} + +\begin{Lemma} + Suppose we have $c, c' \in \Q_p^{|x|}$ such that $f(c), f(c')$ are in the same partition and $g(f(c)) = g(f(c'))$. + Then $c, c'$ have the same $\Phi$-type over $B$. +\end{Lemma} + +\begin{proof} + Let $a_i = \vec p_i \cdot \vec c$ and $a_i' = \vec p_i \cdot \vec c'$ so that + + \begin{align*} + f(c) &= (p_i(c))_{i \in I} = (\vec p_i \cdot \vec c)_{i \in I} = (a_i)_{i \in I} \\ + f(c') &= (p_i(c'))_{i \in I} = (\vec p_i \cdot \vec c')_{i \in I} = (a_i')_{i \in I} + \end{align*} + + For each $i$ we show that $a_i, a_i'$ are in the same subinterval and have the same subinterval type, so the conclusion follows by Lemma \ref{main_lemma} + ($f(c), f(c')$ are in the same partition ensuring the proper order of T-valuations for the 3rd condition of the lemma). + $\It$ records the subinterval type of each element, so if $g(\bar a) = g(\bar a')$ then $a_i, a_i'$ have the same subinterval type for all $i \in I$. + Thus it remains to show that $a_i, a_i'$ lie in the same subinterval for all $i \in I$. + Suppose $i$ is an independent index. + Then by construction, $\Sub$ records the subinterval for $a_i, a_i'$, so those have to belong to the same subinterval. + Now suppose $i$ is dependent. + Pick the largest $j < i$ such that $j$ is independent. + We have $F(a_i) \leq F(a_j)$ and $F(a_i') \leq F(a_j')$. + Moreover $F(a_j) = F(a_j')$ as $a_j, a_j'$ lie in the same subinterval (using the earlier part of the argument as $j$ is independent). + + \begin{Claim} + $\val(a_i - a_i') > F(a_j) - \gamma$ + \end{Claim} + \begin{proof} + Let $K$ be the set of the independent indices less than $i$. + Note that by the definition for dependent indices we have $\vec p_i \in \vecspan \curly{\vec p_k}_{k \in K}$. + We also have + \begin{align*} + \val(a_k - a_k') > F(a_k) \text { for all } k \in K + \end{align*} + as $a_k, a_k'$ lie in the same subinterval (using the earlier part of the argument as $k$ is independent). + \begin{align*} + &\val(a_k - a_k') > F(a_j) \text { for all } k \in K \text{ by monotonicity of $F(a_k)$} \\ + &\val(\vec p_k \cdot \vec c - \vec p_k \cdot \vec c') > F(a_j) \text { for all } k \in K \\ + &\val(\vec p_k \cdot (\vec c - \vec c')) > F(a_j) \text { for all } k \in K \\ + \end{align*} + $K \subset I, i \in I, \vec c - \vec c' \in \Q_p^{|x|}, F(a_j) \in \Z$ + satisfy the requirements of Lemma \ref {gamma}, so we apply it to conclude + \begin{align*} + &\val(\vec p_i \cdot (\vec c - \vec c')) > F(a_j) - \gamma \\ + &\val(\vec p_i \cdot \vec c - \vec p_i \cdot \vec c') > F(a_j) - \gamma \\ + &\val(a_i - a_i') > F(a_j) - \gamma + \end{align*} + as needed, finishing the proof of the claim. + \end{proof} + Additionally $a_i, a_i'$ have the same image in $\Ct$ component, so we have + \begin{align*} + \val(a_i - a_i') > F(a_j) + \end{align*} + We now would like to show that $a_i, a_i'$ lie in the same subinterval. + As $F(a_i) \leq F(a_j)$, $F(a_i') \leq F(a_j')$ and $F(a_j) = F(a_j')$ we have that + $\val(a_i - a_i') > F(a_i)$ and $\val(a_i - a_i') > F(a_i')$ + Suppose that $a_i$ lies in the subinterval $\inti(t, F(a_i), \alpha_U)$ + and that $a_i'$ lies in the subinterval $\inti(t', F(a_i'), \alpha_U')$. + Without loss of generality assume that $F(a_i) \leq F(a_i')$. + As $\val(a_i - a_i') > F(a_i')$, this implies that + \begin{align*} + a_i &\in B(a_i', F(a_i')) \\ + a_i &\in B(t', F(a_i')) \\ + B(t, F(a_i)) &\cap B(t', F(a_i')) \neq \emptyset \\ + B(t, F(a_i)) &\subset B(t', F(a_i')) + \end{align*} + For the subintervals to be disjoint we need + $\inti(t, F(a_i), \alpha_U) \cap B(t', F(a_i')) = \emptyset$. + But $\val(t' - a_i) > F(a_i')$ implying that $a_i \in \inti(t, F(a_i), \alpha_U) \cap B(t', F(a_i'))$ giving a contradiction. + Therefore the subintervals coicide finishing the proof. +\end{proof} + +\begin{Corollary} + $\Phi(x,y)$ has dual $\vc$-density $\leq |x|$. +\end{Corollary} + +\begin{proof} + Suppose we have $c, c' \in \Q_p^{|x|}$ such that $f(c), f(c')$ are in the same partition and $g(f(c)) = g(f(c'))$. + Then by the previous lemma $c, c'$ have the same $\Phi$-type. + Thus the number of possible $\Phi$-types is bounded by the size of the range of $g$ times the number of possible partitions + + \begin{align*} + \text{(number of partitions)} \cdot |\It|^{|I|} \cdot |\Sub|^{|J|} \cdot |\Ct|^{|I-J|} + \end{align*} + + There are at most $\paren{|2I|!}^2$ many partitions of $\Sg$, + so in the product above, the only component dependent on $B$ is + + \begin{align*} + |\Sub|^{|J|} \leq (N \cdot 3{|I|}^2)^{|J|} = O(N^{|J|}) + \end{align*} + + Every $p_i$ is an element of a $|x|$-dimensional vector space, so there can be at most $|x|$ many independent vectors. + Thus we have $|J| \leq |x|$ and the bound follows. +\end{proof} + +\begin{Corollary} [Theorem \ref{main_theorem}] + $\LL_{aff}$-structure $\Q_p$ has $\vc(n) = n$. +\end{Corollary} + +\begin{proof} + The previous lemma implies that $\vc^*(\phi) \leq \vc^*(\Phi) \leq |x|$. + As choice of $\phi$ was arbitrary, this implies that the vc-density of any formula is bounded by the arity of $x$. +\end{proof} + +This proof relies heavily on the linearity of functions $a_1, a_2, c$ in the cell deomposition result (see Definition \ref{cell}). +Linearity is used to separate $x$ and $y$ variables as well as +for Corollary \ref{gamma} to reduce the number of independent factors from $|I|$ to $|x|$. +The paper \cite{reduct} has cell decomposition results for more expressive reducts of $\Q_p$, +including, for exapmple, restricted multiplication. +While our results don't apply to it directly, +it is this author's hope that similar techniques can be used to compute $\vc(n)$ function for those structures. + +\begin{thebibliography}{9} +\bibitem{density} + M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, S. Starchenko, + \textit{Vapnik-Chervonenkis density in some theories without the independence property}, I, + Trans. Amer. Math. Soc. 368 (2016), 5889-5949 +\bibitem{reduct} + E. Leenknegt. \textit{Reducts of $p$-adically closed fields}, Archive for Mathematical Logic, 53(3):285-306, 2014 +\end{thebibliography} + + + + + + + + + + +\end{document} + + + + + + + + diff --git a/research/10 QP reduct/QP_reduct.pdf b/research/10 QP reduct/QP_reduct.pdf index 5b79ae0f..a8bb2310 100644 Binary files a/research/10 QP reduct/QP_reduct.pdf and b/research/10 QP reduct/QP_reduct.pdf differ diff --git a/research/10 QP reduct/QP_reduct.tex b/research/10 QP reduct/QP_reduct.tex index 5a56d5ef..cff017ef 100644 --- a/research/10 QP reduct/QP_reduct.tex +++ b/research/10 QP reduct/QP_reduct.tex @@ -102,169 +102,8 @@ We denote the arity of a tuple $x$ of variables by $|x|$. % First-order formulas will have parameter variables separated $\phi(x; y)$. -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -\section{VC-dimension and vc-density} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - - Throughout this section we work with a collection $\F$ of subsets of a set $X$. - We call the pair $(X, \F)$ a \defn{set system}. - -\begin{Definition} - \begin{itemize} - \item Given a subset $A$ of $X$, we define the set system $(A, A \cap \F)$ - where $A \cap \F = \curly{A \cap F \mid F\in \F}$. - \item For $A \subset X$ we say that $\F$ \defn{shatters} $A$ if $A \cap \F = \PP(A)$ (the power set of $A$). - \end{itemize} -\end{Definition} - -\begin{Definition} - We say $(X, \F)$ has \defn{VC-dimension} $n$ if the largest subset of $X$ shattered by $\F$ is of size $n$. - If $\F$ shatters arbitrarily large subsets of $X$, we say that $(X, \F)$ has infinite VC-dimension. - We denote the VC-dimension of $(X, \F)$ by $\VC(X, \F)$. -\end{Definition} - -\begin{Note} - We may drop $X$ from the $\VC(X, \F)$, as the VC-dimension doesn't depend on the base set and is determined by $(\bigcup \F, \F)$. -\end{Note} -Set systems of finite VC-dimension tend to have good combinatorial properties, -and we consider set systems with infinite VC-dimension to be poorly behaved. - -Another natural combinatorial notion is that of a dual system: -\begin{Definition} - For $a \in X$ define $X_a = \curly{F \in \F \mid a \in F}$. - Let $\F^* = \curly{X_a \mid a \in X}$. - We call $(\F, \F^*)$ the \defn{dual system} of $(X, \F)$. - The VC-dimension of the dual system of $(X, \F)$ is referred to as the \defn{dual VC-dimension} of $(X, \F)$ and denoted by $\VC^*(\F)$. - (As before, this notion doesn't depend on $X$.) -\end{Definition} - -\begin{Lemma} - A set system $(X, \F)$ has finite VC-dimension if and only if its dual system has finite VC-dimension. - More precisely - \begin{align*} - \VC^*(\F) \leq 2^{1+\VC(\F)}. - \end{align*} -\end{Lemma} - -For a more refined notion of complexity of $(X, \F)$ we look at the traces of our family on finite sets: -\begin{Definition} - Define the \defn{shatter function} $\pi_\F \colon \N \arr \N$ and the \defn{dual shatter function} $\pi^*_\F \colon \N \arr \N$ of $\F$ by - \begin{align*} - \pi_\F(n) &= \max \curly{|A \cap \F| \mid A \subset X \text{ and } |A| = n} \\ - \pi^*_\F(n) &= \max \curly{\text{atoms($B$)} \mid B \subset \F, |B| = n} - \end{align*} - where atoms($B$) = number of atoms in the Boolean algebra of sets generated by $B$. - Note that the dual shatter function is precisely the shatter function of the dual system: $\pi^*_\F = \pi_{\F^*}$. -\end{Definition} - -A simple upper bound is $\pi_\F(n) \leq 2^n$ (same for the dual). -If the VC-dimension of $\F$ is infinite then clearly $\pi_\F(n) = 2^n$ for all $n$. Conversely we have the following remarkable fact: -\begin{Theorem} [Sauer-Shelah '72] - If the set system $(X, \F)$ has finite VC-dimension $d$ then $\pi_\F(n) \leq {n \choose \leq d}$ for all $n$, where - ${n \choose \leq d} = {n \choose d} + {n \choose d - 1} + \ldots + {n \choose 1}$. -\end{Theorem} - -Thus the systems with a finite VC-dimension are precisely the systems where the shatter function grows polynomially. -Define the vc-density of $\F$ to quantify the growth of the shatter function of $\F$: -\begin{Definition} - Define \defn{vc-density} and \defn{dual vc-density} of $\F$ as - \begin{align*} - \vc(\F) &= \limsup_{n \to \infty}\frac{\log \pi_\F(n)}{\log n} \in \R^{\geq 0} \cup \curly{+\infty},\\ - \vc^*(\F) &= \limsup_{n \to \infty}\frac{\log \pi^*_\F(n)}{\log n}\in \R^{\geq 0} \cup \curly{+\infty}. - \end{align*} -\end{Definition} - -Generally speaking a shatter function that is bounded by a polynomial doesn't itself have to be a polynomial. -Proposition 4.12 in \cite{density} gives an example of a shatter function that grows like $n \log n$ (so it has vc-density $1$). - -So far the notions that we have defined are purely combinatorial. -We now adapt VC-dimension and vc-density to the model theoretic context. - -\begin{Definition} - Work in a first-order structure $M$. - Fix a finite collection of formulas $\Phi(x, y)$. - - \begin{itemize} - \item For $\phi(x, y) \in \LL(M)$ and $b \in M^{|y|}$ let - \begin{align*} - \phi(M^{|x|}, b) = \{a \in M^{|x|} \mid \phi(a, b)\} \subseteq M^{|x|}. - \end{align*} - \item Let $\Phi(M^{|x|}, M^{|y|})= \{\phi(M^{|x|}, b) \mid \phi_i \in \Phi, b \in M^{|y|}\} \subseteq \PP(M^{|x|})$. - \item Let $\F_\Phi = \Phi(M^{|x|}, M^{|y|})$, giving rise to a set system $(M^{|x|}, \F_\Phi)$. - \item Define the \defn{VC-dimension} of $\Phi$, $\VC(\Phi)$ to be the VC-dimension of $(M^{|x|}, \F_\Phi)$, similarly for the dual. - \item Define the \defn{vc-density} of $\Phi$, $\vc(\Phi)$ to be the vc-density of $(M^{|x|}, \F_\Phi)$, similarly for the dual. - \end{itemize} - - We will also refer to the vc-density and VC-dimension of a single formula $\phi$ - viewing it as a one element collection $\Phi = \curly{\phi}$. -\end{Definition} - -Counting atoms of a Boolean algebra in a model theoretic setting corresponds to counting types, -so it is instructive to rewrite the shatter function in terms of types. - -\begin{Definition} - \begin{align*} - \pi^*_\Phi(n) &= \max \curly{\text{number of $\Phi$-types over $B$} \mid B \subset M, |B| = n} - \end{align*} - Here a $\Phi$-type over $B$ is a maximal consistent collection of functions of the form $\phi(x, b)$ or $\neg\phi(x, b)$ - where $\phi \in \Phi$ and $b \in B$. -\end{Definition} - -\begin{Lemma} \label{count_types} - \begin{align*} - \vc^*(\Phi) &= \text{degree of polynomial growth of $\pi^*_\Phi(n)$} = \limsup_{n \to \infty}\frac{\log \pi^*_\Phi(n)}{\log n} - \end{align*} -\end{Lemma} - -\begin{proof} - \begin{align*} - &\pi^*_{\F_\Phi}\paren{n} \leq \pi^*_\Phi(n) \leq \pi^*_{\F_\Phi}\paren{|\Phi|n} \\ - &\vc^*(\Phi) \leq \limsup_{n \to \infty}\frac{\log \pi^*_\Phi(n)}{\log n} \leq \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{|\Phi|n}}{\log n} = \\ - & = \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{|\Phi|n}}{\log |\Phi|n} \frac{\log |\Phi|n}{\log n} = - \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{|\Phi|n}}{\log |\Phi|n} \leq \\ - &\leq \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{n}}{\log n} = \vc^*(\Phi) - \end{align*} -\end{proof} - -One can check that the shatter function and hence VC-dimension and vc-density of a formula are elementary notions, -so they only depend on the first-order theory of the structure. - -NIP theories are a natural context for studying vc-density. -In fact we can take the following as the definition of NIP: -\begin{Definition} - Define $\phi$ to be NIP if it has finite VC-dimension. - A theory $T$ is NIP if all the formulas are NIP. -\end{Definition} - -% \cite{Aschenbrenner_reference_8} shows that in a -% \item general combinatorial context, -In a general combinatorial context for arbitrary set systems, -vc-density can be any real number in $0 \cup [1, \infty)$. -Less is known if we restrict our attention to NIP theories. -Proposition 4.6 in \cite{density} gives examples of formulas that have non-integer rational vc-density in an NIP theory, -however it is open whether one can get an irrational vc-density in this model-theoretic setting. - -Instead of working with a theory formula by formula, we can look for a uniform bound for all formulas: -\begin{Definition} \label{vc_fn_def} - For a given NIP structure $M$, define the \defn{vc-function} - \begin{align*} - \vc^M(n) &= \sup \{\vc^*(\phi(x, y)) \mid \phi \in \LL(M), |x| = n\} \\ - &= \sup \{\vc(\phi(x, y)) \mid \phi \in \LL(M), |y| = n\} - \end{align*} -\end{Definition} - -As before this definition is elementary, so it only depends on the theory of $M$. -We omit the superscript $M$ if it is understood from the context. -One can easily check the following bounds: -\begin{Lemma} [Lemma 3.22 in \cite{density}] We have $\vc(1) \geq 1$ and $\vc(n) \geq n\vc(1)$. - -\end{Lemma} +\input{../vc_intro.tex} -However, it is not known whether the second inequality can be strict or even whether $\vc(1) < \infty$ implies $\vc(n) < \infty$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% diff --git a/research/vc_intro.tex b/research/vc_intro.tex new file mode 100644 index 00000000..7e2a5100 --- /dev/null +++ b/research/vc_intro.tex @@ -0,0 +1,164 @@ + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\section{VC-dimension and vc-density} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + +Throughout this section we work with a collection $\F$ of subsets of a set $X$. +We call the pair $(X, \F)$ a \defn{set system}. + +\begin{Definition} + \begin{itemize} \ \\ + \item Given a subset $A$ of $X$, we define the set system $(A, A \cap \F)$ + where $A \cap \F = \curly{A \cap F \mid F\in \F}$. + \item For $A \subset X$ we say that $\F$ \defn{shatters} $A$ if $A \cap \F = \PP(A)$ (the power set of $A$). + \end{itemize} +\end{Definition} + +\begin{Definition} + We say $(X, \F)$ has \defn{VC-dimension} $n$ if the largest subset of $X$ shattered by $\F$ is of size $n$. + If $\F$ shatters arbitrarily large subsets of $X$, we say that $(X, \F)$ has infinite VC-dimension. + We denote the VC-dimension of $(X, \F)$ by $\VC(X, \F)$. +\end{Definition} + +\begin{Note} + We may drop $X$ from the $\VC(X, \F)$, as the VC-dimension doesn't depend on the base set and is determined by $(\bigcup \F, \F)$. +\end{Note} +Set systems of finite VC-dimension tend to have good combinatorial properties, +and we consider set systems with infinite VC-dimension to be poorly behaved. + +Another natural combinatorial notion is that of a dual system: +\begin{Definition} + For $a \in X$ define $X_a = \curly{F \in \F \mid a \in F}$. + Let $\F^* = \curly{X_a \mid a \in X}$. + We call $(\F, \F^*)$ the \defn{dual system} of $(X, \F)$. + The VC-dimension of the dual system of $(X, \F)$ is referred to as the \defn{dual VC-dimension} of $(X, \F)$ and denoted by $\VC^*(\F)$. + (As before, this notion doesn't depend on $X$.) +\end{Definition} + +\begin{Lemma} + A set system $(X, \F)$ has finite VC-dimension if and only if its dual system has finite VC-dimension. + More precisely + \begin{align*} + \VC^*(\F) \leq 2^{1+\VC(\F)}. + \end{align*} +\end{Lemma} + +For a more refined notion of complexity of $(X, \F)$ we look at the traces of our family on finite sets: +\begin{Definition} + Define the \defn{shatter function} $\pi_\F \colon \N \arr \N$ and the \defn{dual shatter function} $\pi^*_\F \colon \N \arr \N$ of $\F$ by + \begin{align*} + \pi_\F(n) &= \max \curly{|A \cap \F| \mid A \subset X \text{ and } |A| = n} \\ + \pi^*_\F(n) &= \max \curly{\text{atoms($B$)} \mid B \subset \F, |B| = n} + \end{align*} + where atoms($B$) = number of atoms in the Boolean algebra of sets generated by $B$. + Note that the dual shatter function is precisely the shatter function of the dual system: $\pi^*_\F = \pi_{\F^*}$. +\end{Definition} + +A simple upper bound is $\pi_\F(n) \leq 2^n$ (same for the dual). +If the VC-dimension of $\F$ is infinite then clearly $\pi_\F(n) = 2^n$ for all $n$. Conversely we have the following remarkable fact: +\begin{Theorem} [Sauer-Shelah '72] + If the set system $(X, \F)$ has finite VC-dimension $d$ then $\pi_\F(n) \leq {n \choose \leq d}$ for all $n$, where + ${n \choose \leq d} = {n \choose d} + {n \choose d - 1} + \ldots + {n \choose 1}$. +\end{Theorem} + +Thus the systems with a finite VC-dimension are precisely the systems where the shatter function grows polynomially. +Define the vc-density of $\F$ to quantify the growth of the shatter function of $\F$: +\begin{Definition} + Define \defn{vc-density} and \defn{dual vc-density} of $\F$ as + \begin{align*} + \vc(\F) &= \limsup_{n \to \infty}\frac{\log \pi_\F(n)}{\log n} \in \R^{\geq 0} \cup \curly{+\infty},\\ + \vc^*(\F) &= \limsup_{n \to \infty}\frac{\log \pi^*_\F(n)}{\log n}\in \R^{\geq 0} \cup \curly{+\infty}. + \end{align*} +\end{Definition} + +Generally speaking a shatter function that is bounded by a polynomial doesn't itself have to be a polynomial. +Proposition 4.12 in \cite{density} gives an example of a shatter function that grows like $n \log n$ (so it has vc-density $1$). + +So far the notions that we have defined are purely combinatorial. +We now adapt VC-dimension and vc-density to the model theoretic context. + +\begin{Definition} + Work in a first-order structure $M$. + Fix a finite collection of formulas $\Phi(x, y)$. + + \begin{itemize} + \item For $\phi(x, y) \in \LL(M)$ and $b \in M^{|y|}$ let + \begin{align*} + \phi(M^{|x|}, b) = \{a \in M^{|x|} \mid \phi(a, b)\} \subseteq M^{|x|}. + \end{align*} + \item Let $\Phi(M^{|x|}, M^{|y|})= \{\phi(M^{|x|}, b) \mid \phi_i \in \Phi, b \in M^{|y|}\} \subseteq \PP(M^{|x|})$. + \item Let $\F_\Phi = \Phi(M^{|x|}, M^{|y|})$, giving rise to a set system $(M^{|x|}, \F_\Phi)$. + \item Define the \defn{VC-dimension} of $\Phi$, $\VC(\Phi)$ to be the VC-dimension of $(M^{|x|}, \F_\Phi)$, similarly for the dual. + \item Define the \defn{vc-density} of $\Phi$, $\vc(\Phi)$ to be the vc-density of $(M^{|x|}, \F_\Phi)$, similarly for the dual. + \end{itemize} + + We will also refer to the vc-density and VC-dimension of a single formula $\phi$ + viewing it as a one element collection $\Phi = \curly{\phi}$. +\end{Definition} + +Counting atoms of a Boolean algebra in a model theoretic setting corresponds to counting types, +so it is instructive to rewrite the shatter function in terms of types. + +\begin{Definition} + \begin{align*} + \pi^*_\Phi(n) &= \max \curly{\text{number of $\Phi$-types over $B$} \mid B \subset M, |B| = n} + \end{align*} + Here a $\Phi$-type over $B$ is a maximal consistent collection of functions of the form $\phi(x, b)$ or $\neg\phi(x, b)$ + where $\phi \in \Phi$ and $b \in B$. +\end{Definition} + +\begin{Lemma} \label{count_types} + \begin{align*} + \vc^*(\Phi) &= \text{degree of polynomial growth of $\pi^*_\Phi(n)$} = \limsup_{n \to \infty}\frac{\log \pi^*_\Phi(n)}{\log n} + \end{align*} +\end{Lemma} + +\begin{proof} + \begin{align*} + &\pi^*_{\F_\Phi}\paren{n} \leq \pi^*_\Phi(n) \leq \pi^*_{\F_\Phi}\paren{|\Phi|n} \\ + &\vc^*(\Phi) \leq \limsup_{n \to \infty}\frac{\log \pi^*_\Phi(n)}{\log n} \leq \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{|\Phi|n}}{\log n} = \\ + & = \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{|\Phi|n}}{\log |\Phi|n} \frac{\log |\Phi|n}{\log n} = + \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{|\Phi|n}}{\log |\Phi|n} \leq \\ + &\leq \limsup_{n \to \infty}\frac{\log \pi^*_{\F_\Phi}\paren{n}}{\log n} = \vc^*(\Phi) + \end{align*} +\end{proof} + +One can check that the shatter function and hence VC-dimension and vc-density of a formula are elementary notions, +so they only depend on the first-order theory of the structure. + +NIP theories are a natural context for studying vc-density. +In fact we can take the following as the definition of NIP: +\begin{Definition} + Define $\phi$ to be NIP if it has finite VC-dimension. + A theory $T$ is NIP if all the formulas are NIP. +\end{Definition} + +% \cite{Aschenbrenner_reference_8} shows that in a +% \item general combinatorial context, +In a general combinatorial context for arbitrary set systems, +vc-density can be any real number in $0 \cup [1, \infty)$. +Less is known if we restrict our attention to NIP theories. +Proposition 4.6 in \cite{density} gives examples of formulas that have non-integer rational vc-density in an NIP theory, +however it is open whether one can get an irrational vc-density in this model-theoretic setting. + +Instead of working with a theory formula by formula, we can look for a uniform bound for all formulas: +\begin{Definition} \label{vc_fn_def} + For a given NIP structure $M$, define the \defn{vc-function} + \begin{align*} + \vc^M(n) &= \sup \{\vc^*(\phi(x, y)) \mid \phi \in \LL(M), |x| = n\} \\ + &= \sup \{\vc(\phi(x, y)) \mid \phi \in \LL(M), |y| = n\} + \end{align*} +\end{Definition} + +As before this definition is elementary, so it only depends on the theory of $M$. +We omit the superscript $M$ if it is understood from the context. +One can easily check the following bounds: +\begin{Lemma} [Lemma 3.22 in \cite{density}] We have $\vc(1) \geq 1$ and $\vc(n) \geq n\vc(1)$. + +\end{Lemma} + +However, it is not known whether the second inequality can be strict or even whether $\vc(1) < \infty$ implies $\vc(n) < \infty$.