arXiv subjects

C. Terry

Publications and source records attributed to C. Terry.

At least 19 recordsLinked to original sources

arXiv2026

Quantitative analytic stable regularity

We prove quantitative stable regularity lemmas for binary real-valued functions, extending the work of Malliaris and Shelah for stable graphs. The statements of our results are modeled after non-quantitative theorems for stable functions due to Chavarria, Conant, and Pillay. One of the key tools in our quantitative proof is an "analytic symmetry lemma", which gives a function-theoretic analogue of the fact that a pair of good sets in a graph has density close to 0 or 1. We also develop a function-theoretic treatment of Malliaris and Shelah's random sampling method for refining partitions consisting of good sets into equipartitions.

math.LO↗

arXiv2025

On the quadratic complexity of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC_{2}}$-dimension

In prior work, we showed that subsets of $\mathbb{F}_{p}^{n}$ of $\mathrm{VC_{2}}$-dimension at most $k$ are well approximated by a union of atoms of a quadratic factor of complexity $(\ell,q)$, where the complexity $\ell$ of the linear part and the complexity $q$ of the quadratic part are both bounded in terms of $k$, $p$, and the desired level of approximation $\mu$. A key tool in the proof of this result was an arithmetic regularity lemma for the Gowers $U^3$-norm by Green and Tao, which resulted in tower-type bounds (in terms of $\mu^{-1}$) on both $\ell$ and $q$. In the present paper we show that for sets of bounded $\mathrm{VC}_2$-dimension, the bound on $q$ can be substantially improved. Specifically, we will prove that any set $A\subseteq G=\mathbb{F}_p^n$ of $\mathrm{VC}_2$-dimension at most $k$ is approximately equal (up to error $\mu |G|$) to a union of atoms of a quadratic factor whose quadratic complexity is at most $\log_p(\mu^{-k-o(1)})$, implying that the purely quadratic component of the factor partitions the group into $\mu^{-k-o(1)}$ many parts. We achieve this by using our earlier result to obtain an initial quadratic factor $\mathcal{B}$, and then applying a generalization of an argument of Alon, Fox and Zhao for subsets of $\mathbb{F}_{p}^{n}$ of bounded $\mathrm{VC}$-dimension to the label space (also known as "configuration space") of $\mathcal{B}$. A related strategy was employed in earlier work of the authors on $\mathrm{NFOP}_2$ subsets of $\mathbb{F}_p^n$, and in work of the first author in the context of 3-uniform hypergraphs.

math.CO↗

arXiv2025

The structure of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC}_2$-dimension

We show that a subset of $\mathbb{F}_{p}^{n}$ of $\mathrm{VC_{2}}$-dimension at most $k$ is well approximated by a union of atoms of a quadratic factor of complexity $(\ell,q)$ (denoting the complexities of the linear and quadratic part, respectively), where $\ell$ and $q$ are bounded by a constant depending only on $k$ and the desired level of approximation. This generalises a result of Alon, Fox and Zhao on the structure of sets of bounded $\mathrm{VC}$-dimension, and is analogous to contemporaneous work of the authors arXiv:2111.01737 in the setting of 3-uniform hypergraphs. The main result originally appeared--albeit with a different proof--in a 2021 preprint arXiv:2111.01739, which has since been split into two: the present work, which focuses on higher arity NIP and develops a theory of local uniformity semi-norms of possibly independent interest, and its companion arXiv:2111.01739, which strengthens these results under a generalized notion of stability.

math.CO↗

arXiv2025

Stabilizers and NIP arithmetic regularity

We give a new proof of the NIP arithmetic regularity lemma for finite groups (due to the authors and Pillay), which describes the approximate structure of "NIP sets" in finite groups, i.e., subsets whose collection of left translates has bounded VC-dimension. Our new proof avoids sophisticated ingredients from the model theory of NIP formulas (e.g., Borel definability and generic compact domination). The key tool is an elaboration on an elementary lemma due to Alon, Fox, and Zhao concerning the behavior of subgroups contained in stabilizers. We adapt this lemma to arbitrary subsets of stabilizers using technical (but elementary) maneuvers based on work of Sisask. Using another trick from Alon, Fox, and Zhao, we then give an effective proof of a related result of the first author and Pillay on finite NIP sets of bounded tripling in arbitrary groups. Along the way, we show that NIP sets satisfy a strong form of the Polynomial Bogolyubov-Ruzsa Conjecture.

math.CO↗

arXiv2024

Growth of regular partitions 3: strong regularity and the vertex partition

We consider here the strong regularity for $3$-uniform hypergraphs developed by Frankl, Gowers, Kohayakawa, Nagle, R\"{o}dl, Skokan, and Schacht. This type of regular decomposition comes with two components, a partition of the vertices, and a partition of the pairs of vertices. The data of a regular decomposition also includes two parameters measuring quasirandomness, a fixed constant $\epsilon_1>0$, and a function $\epsilon_2:\mathbb{N}\rightarrow (0,1]$. We define two growth functions associated to a hereditary property $\mathcal{H}$ of $3$-uniform hypergraphs: $T_{\mathcal{H}}(\epsilon_1,\epsilon_2)$ which measures the size of the vertex component, and $L_{\mathcal{H}}(\epsilon,\epsilon_2)$ which measures the size of the pairs component. We introduce the following question. What are the possible asymptotic growth rates of functions of the form $T_{\mathcal{H}}$ and $L_{\mathcal{H}}$? In this paper, we consider this question for $T_{\mathcal{H}}$, proving a separation into four classes: constant, polynomial, exponential, or at least wowzer. The separations among the constant, polynomial and exponential ranges require only slow growing (namely polynomial) choices for $\epsilon_2$. The jump to the wowzer range uses a very fast growing $\epsilon_2$ and makes crucial use of a lower bound construction for strong graph regularity due to Conlon and Fox.

math.CO↗

arXiv2024

Growth of regular partitions 4: strong regularity and the pairs partition

This paper studies bounds in a strong form of regularity for $3$-uniform hypergraphs which was developed by Frankl, Gowers, Kohayakawa, Nagle, R\"{o}dl, Skokan, and Schacht. Regular decompositions of this type involve two structural components: a partition on the vertex set and a partition on the pairs of vertices. The regularity of such decompositions are measured by two parameters: an $\epsilon_1>0$ and a function $\epsilon_2:\mathbb{N}\rightarrow (0,1]$. To each hereditary property $\mathcal{H}$ of $3$-uniform hypergraphs, we associate two corresponding growth functions: $T_{\mathcal{H}}(\epsilon_1,\epsilon_2)$ for the size of the vertex component, and $L_{\mathcal{H}}(\epsilon_1,\epsilon_2)$ for the size of the pairs component. The problem of understanding the asymptotic growth of such functions was introduced in a companion paper, which also proved several results about $T_{\mathcal{H}}$. In this paper we study the possible asymptotic behavior of $L_{\mathcal{H}}$. We show any such function is either constant, bounded above and below by a polynomial, or bounded below by an exponential. All results require only reasonable growth rates for $\epsilon_2$ (namely polynomial).

math.CO↗

arXiv2024

Growth of regular partitions 1: Improved bounds for small slicewise VC-dimension

This is Part 1 in a series of papers about sizes of regular partitions of $3$-uniform hypergraphs. Previous work of the author and Wolf, and independently Chernikov and Towsner, showed that $3$-uniform hypergraphs of small slicewise VC-dimension admit homogeneous partitions. The results of Chernikov and Towsner did not produce explicit bounds, while the work of the author and Wolf relied on a strong version of hypergraph regularity, and consequently produced a Wowzer type bound on the size of the partition. This paper gives a new proof of this result, yielding $\e$-homogeneous partitions of size at most $2^{2^{\epsilon^{-K}}}$, where $K$ is a constant depending on the slicewise VC-dimension. This result is a crucial ingredient in Part 2 of the series, which investigates the growth of weak regular partitions in hereditary properties of $3$-uniform hypergraphs.

math.CO↗

arXiv2024

Growth of regular partitions 2: Weak regularity

This is Part 2 in a series of papers about the growth of regular partitions in hereditary properties $3$-uniform hypergraphs. The focus of this paper is the notion of weak hypergraph regularity, first developed by Chung, Chung-Graham, and Haviland-Thomason. Given a hereditary property of $3$-uniform hypergraphs $\mathcal{H}$, we define a function $M_{\mathcal{H}}:(0,1)\rightarrow \mathbb{N}$ by letting $M_{\mathcal{H}}(\epsilon)$ be the smallest integer $M$ such that all sufficiently large elements of $\mathcal{H}$ admit weak regular partitions of size at most $M$. We show the asymptotic growth rate of such a function falls into one of four categories: constant, polynomial, between single and double exponentials, or tower. These results are a crucial component in Part 3 of the series, which considers vertex partitions associated to a stronger notion of hypergraph regularity.

math.CO↗

arXiv2023

Pseudofinite proofs of the stable graph regularity lemma

This expository article is based on two lectures given by the first author at the Fields Institute in the Fall 2021 Thematic Program on Trends in Pure and Applied Model Theory. We give a detailed proof of a qualitative version of the Mallaris-Shelah regularity lemma for stable graphs using only basic local stability theory and an ultraproduct construction. This proof strategy was first established by Malliaris and Pillay, and later simplified by Pillay. We provide some further simplifications, and also explain how the pseudofinite approach can be used to obtain a qualitative strengthening (compared to previous proofs) in terms of "functional error". To illustrate the extra leverage obtained by functional error, we give an elementary argument for extracting equipartitions from arbitrary partitions.

math.LO↗

arXiv2023

Higher arity stability and the functional order property

The $k$-dimensional functional order property ($\text{FOP}_k$) is a combinatorial property of a $(k+1)$-partitioned formula. This notion arose in work of Terry and Wolf, which identified $\text{NFOP}_2$ as a ternary analogue of stability in the context of two finitary combinatorial problems related to hypergraph regularity and arithmetic regularity. In this paper we show $\text{NFOP}_k$ has equally strong implications in model-theoretic classification theory, where its behavior as a $(k+1)$-ary version of stability is in close analogy to the behavior of $k$-dependence as a $(k+1)$-ary version of $\text{NIP}$. Our results include several new characterizations of $\text{NFOP}_k$, including a characterization in terms of collapsing indiscernibles, combinatorial recharacterizations, and a characterization in terms of type-counting when $k=2$. As a corollary of our collapsing theorem, we show $\text{NFOP}_k$ is closed under Boolean combinations, and that $\text{FOP}_k$ can always be witnessed by a formula where all but one variable have length $1$. When $k=2$, we prove a composition lemma analogous to that of Chernikov and Hempel from the setting of $2$-dependence. Using this, we provide a new class of algebraic examples of $\text{NFOP}_2$ theories. Specifically, we show that if $T$ is the theory of an infinite dimensional vector space over a field $K$, equipped with a bilinear form satisfying certain properties, then $T$ is $\text{NFOP}_2$ if and only if $K$ is stable. Along the way we provide a corrected and reorganized proof of Granger's quantifier elimination and completeness results for these theories.

math.LO↗

arXiv2022

An improved bound for regular decompositions of $3$-uniform hypergraphs of bounded $VC_2$-dimension

A regular partition $\mathcal{P}$ for a $3$-uniform hypergraph $H=(V,E)$ consists of a partition $V=V_1\cup \ldots \cup V_t$ and for each $ij\in {[t]\choose 2}$, a partition $K_2[V_i,V_j]=P_{ij}^1\cup \ldots \cup P_{ij}^{\ell}$, such that certain quasirandomness properties hold. The complexity of $\mathcal{P}$ is the pair $(t,\ell)$. In this paper we show that if a $3$-uniform hypergraph $H$ has $VC_2$-dimension at most $k$, then there is a regular partition $\mathcal{P}$ for $H$ of complexity $(t,\ell)$, where $\ell$ is bounded by a polynomial in the degree of regularity. This is a vast improvement on the bound arising from the proof of this regularity lemma in general, in which the bound generated for $\ell$ is of Wowzer type. This can be seen as a higher arity analogue of the efficient regularity lemmas for graphs and hypergraphs of bounded VC-dimension due to Alon-Fischer-Newman, Lov\'{a}sz-Szegedy, and Fox-Pach-Suk.

math.CO↗

arXiv2021

Irregular triads in 3-uniform hypergraphs

Over the past several years, numerous authors have explored model theoretically motivated combinatorial conditions that ensure that a graph has an efficient regular decomposition in the sense of Szemer\'edi. In this paper we set out a research program that explores a corresponding set of questions for 3-uniform hypergraphs, a setting in which useful notions of regularity are significantly more intricate. The main results in this paper concern certain combinatorial properties which arose as natural higher-order generalizations of the order property in parallel work of the authors in the arithmetic setting. Interpreted in the context of 3-uniform hypergraphs, these are tightly connected to the nature of irregular triads. Specifically, we show that a hereditary property of 3-uniform hypergraphs admits regular decompositions with so-called "linear error" if and only if it does not have the functional order property. Along the way, we show that a hereditary property of 3-uniform hypergraphs is homogeneous (i.e. all regular triads have density near $0$ or near $1$) if and only it has bounded $\textrm{VC}_2$-dimension. This provides a quantitative version of a recent result of Chernikov and Towsner. We also address several questions arising from prior work on tame regularity in hypergraphs. In particular, we characterize the hereditary properties of $3$-uniform hypergraphs admitting the type of regular partitions appearing in work of Fox et al. as those that have bounded slicewise $\textrm{VC}$-dimension. This is again analogous to a recent non-quantitative result of Chernikov and Towsner.

math.CO↗

arXiv2021

Higher-order generalizations of stability and arithmetic regularity

We define a natural notion of higher order stability and show that subsets of $\mathbb{F}_p^n$ that are tame in this sense can be approximately described by a union of low-complexity quadratic varieties, up to linear error. This generalizes the arithmetic regularity lemma for stable subsets of $\mathbb{F}_p^n$, proved in earlier work of the authors, to the realm of higher-order Fourier analysis. This result is strictly stronger than the structure theorem for sets of bounded $\mathrm{VC}_2$-dimension, first proved by the authors in earlier versions of this paper and now available as a separate manuscript arXiv:2510.12867. Taken together, these results provide group theoretic analogues of results obtained for 3-uniform hypergraphs in arXiv:2111.01737.

math.CO↗

arXiv2018

Quantitative structure of stable sets in finite abelian groups

We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A qualitative version of this generalisation was recently obtained by the first author in joint work with Conant and Pillay, using model-theoretic techniques. In contrast, the approach in the present paper is highly quantitative and relies on several key ingredients from arithmetic combinatorics.

math.LO↗

arXiv2018

Structure and regularity for subsets of groups with finite VC-dimension

Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen $\epsilon>0$, describe the structure of $A$ and behave regularly with respect to translates of $A$. For the subclass of groups with uniformly fixed finite exponent $r$, these algebraic objects are normal subgroups with index bounded in terms of $k$, $r$, and $\epsilon$. For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model theoretic, and heavily rely on a structural analysis of compactifications of pseudofinite groups as inverse limits of Lie groups. The introduction of Bohr neighborhoods into the nonabelian setting uses model theoretic methods related to the work of Breuillard, Green, and Tao and Hrushovski on approximate groups, as well as a result of Alekseev, Glebskii, and Gordon on approximate homomorphisms.

math.CO↗

arXiv2017

A group version of stable regularity

We prove that, given $\epsilon>0$ and $k\geq 1$, there is an integer $n$ such that the following holds. Suppose $G$ is a finite group and $A\subseteq G$ is $k$-stable. Then there is a normal subgroup $H\leq G$ of index at most $n$, and a set $Y\subseteq G$, which is a union of cosets of $H$, such that $|A\vartriangle Y|\leq\epsilon|H|$. It follows that, for any coset $C$ of $H$, either $|C\cap A|\leq \epsilon|H|$ or $|C\setminus A|\leq \epsilon|H|$. This qualitatively generalizes recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$.

math.LO↗

arXiv2017

Stable arithmetic regularity in the finite-field model

The arithmetic regularity lemma for $\mathbb{F}_p^n$, proved by Green in 2005, states that given a subset $A\subseteq \mathbb{F}_p^n$, there exists a subspace $H\leq \mathbb{F}_p^n$ of bounded codimension such that $A$ is Fourier-uniform with respect to almost all cosets of $H$. It is known that in general, the growth of the codimension of $H$ is required to be of tower type depending on the degree of uniformity, and that one must allow for a small number of non-uniform cosets. Our main result is that, under a natural model-theoretic assumption of stability, the tower-type bound and non-uniform cosets in the arithmetic regularity lemma are not necessary. Specifically, we prove an arithmetic regularity lemma for $k$-stable subsets $A\subseteq \mathbb{F}_p^n$ in which the bound on the codimension of the subspace is a polynomial (depending on $k$) in the degree of uniformity, and in which there are no non-uniform cosets. This result is an arithmetic analogue of the stable graph regularity lemma proved by Malliaris and Shelah.

math.LO↗

arXiv2017

$\text{VC}_{\ell}$-dimension and the jump to the fastest speed of a hereditary $\mathcal{L}$-property

In this paper we investigate a connection between the growth rates of certain classes of finite structures and a generalization of $\text{VC}$-dimension called $\text{VC}_{\ell}$-dimension. Let $\mathcal{L}$ be a finite relational language with maximum arity $r$. A hereditary $\mathcal{L}$-property is a class of finite $\mathcal{L}$-structures closed under isomorphism and substructures. The \emph{speed} of a hereditary $\mathcal{L}$-property $\mathcal{H}$ is the function which sends $n$ to $|\mathcal{H}_n|$, where $\mathcal{H}_n$ is the set of elements of $\mathcal{H}$ with universe $\{1,\ldots, n\}$. It was previously known there exists a gap between the fastest possible speed of a hereditary $\mathcal{L}$-property and all lower speeds, namely between the speeds $2^{\Theta(n^r)}$ and $2^{o(n^r)}$. We strengthen this gap by showing that for any hereditary $\mathcal{L}$-property $\mathcal{H}$, either $|\mathcal{H}_n|=2^{\Theta(n^r)}$ or there is $\epsilon>0$ such that for all large enough $n$, $|\mathcal{H}_n|\leq 2^{n^{r-\epsilon}}$. This improves what was previously known about this gap when $r\geq 3$. Further, we show this gap can be characterized in terms of $\text{VC}_{\ell}$-dimension, therefore drawing a connection between this finite counting problem and the model theoretic dividing line known as $\ell$-dependence.

math.LO↗