Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

C. Terry, J. Wolf

TL;DR

The paper develops a higher-order structural theory for subsets of the finite vector space $\\mathbb{F}_p^n$ with bounded $\\mathrm{VC}_2$-dimension. By extending quadratic Fourier analysis through local Gowers norms and local IP-operators, it proves a quadratic regularity lemma and a local-uniformity framework that yield a precise structure theorem: sets of bounded $\\mathrm{VC}_2$-dimension are well approximated (up to a small error) by unions of atoms of a high-rank, bounded-complexity quadratic factor. The work also clarifies the behavior of quadratic atoms, showing bounded $\\mathrm{VC}_2$-dimension but potentially large $\\mathrm{VC}$-dimension, and demonstrates that local uniformity on most atoms suffices to capture global density structure. These results generalize linear VC-dimension structure theorems and connect to higher-arity stability and hypergraph regularity, providing a robust framework for higher-order combinatorial decompositions with potential model-theoretic interpretations.

Abstract

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.

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

TL;DR

The paper develops a higher-order structural theory for subsets of the finite vector space with bounded -dimension. By extending quadratic Fourier analysis through local Gowers norms and local IP-operators, it proves a quadratic regularity lemma and a local-uniformity framework that yield a precise structure theorem: sets of bounded -dimension are well approximated (up to a small error) by unions of atoms of a high-rank, bounded-complexity quadratic factor. The work also clarifies the behavior of quadratic atoms, showing bounded -dimension but potentially large -dimension, and demonstrates that local uniformity on most atoms suffices to capture global density structure. These results generalize linear VC-dimension structure theorems and connect to higher-arity stability and hypergraph regularity, providing a robust framework for higher-order combinatorial decompositions with potential model-theoretic interpretations.

Abstract

We show that a subset of of -dimension at most is well approximated by a union of atoms of a quadratic factor of complexity (denoting the complexities of the linear and quadratic part, respectively), where and are bounded by a constant depending only on and the desired level of approximation. This generalises a result of Alon, Fox and Zhao on the structure of sets of bounded -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.
Paper Structure (17 sections, 42 theorems, 229 equations)

This paper contains 17 sections, 42 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Background on quadratic Fourier Analysis
  3. The Gowers $U^2$ norm and the IP operator
  4. An arithmetic regularity lemma for the $U^2$ norm
  5. The Gowers $U^3$ norm and the $\mathrm{IP}_2$-operator
  6. A quadratic arithmetic regularity lemma
  7. The local Gowers semi-norms and local independence property operators
  8. The local $U^2$ semi-norm and the local $\mathrm{IP}$-operator
  9. Interlude: Properties of quadratic atoms
  10. The local $U^3$ semi-norm and the local $\mathrm{IP}_2$-operator
  11. Proving the structure theorem for sets of bounded $\mathrm{VC}_2$-dimension
  12. Preparing the quadratic regularity lemma
  13. Proof of Theorem \ref{['cor:vc2']}
  14. Bilinear exponential sum estimates
  15. Proof of Proposition \ref{['prop:ip2loccontrol']}
  16. ...and 2 more sections

Key Result

Theorem 1.2

For all primes $p$, integers $k\geq 1$, and reals $\varepsilon>0$, there exists $M=M(p,k,\varepsilon)$ such that the following holds. Let $n\geq 1$ be an integer. Suppose that $A\subseteq \mathbb{F}_p^n$ satisfies $\mathrm{VC}(A)<k$. Then for all $\varepsilon>0$, there exists a subgroup $H\leq \math

Theorems & Definitions (82)

  • Definition 1.1: $\mathrm{VC}$-dimension of a subset of a group
  • Theorem 1.2: Arithmetic regularity lemma for subsets of bounded $\mathrm{VC}$-dimension
  • Lemma 1.3: Unions of cosets have bounded $\mathrm{VC}$-dimension
  • proof
  • Corollary 1.4: Structure theorem for sets of bounded $\mathrm{VC}$-dimension
  • Definition 1.5: Stability of a subset of a group
  • Theorem 1.6: Arithmetic regularity lemma for stable sets
  • Definition 1.8: $\mathrm{VC}_2$-dimension of a subset of a group
  • Theorem 1.9: Arithmetic regularity lemma for sets of bounded $\mathrm{VC}_2$-dimension
  • Corollary 1.10: Structure theorem for sets of bounded $\mathrm{VC}_2$-dimension
  • ...and 72 more