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On the linear complexity of subsets of $\mathbb{F}_p^n$ bounded $\textrm{VC}_2$-dimension

Hannah Sheats, Caroline Terry

TL;DR

This work advances our understanding of the structure of subsets of $\mathbb{F}_p^n$ with bounded $\mathrm{VC}_2$-dimension by introducing a cylinder-style quadratic arithmetic regularity lemma and a refined local $U^3$-norm framework. The authors connect localized inverse theorems with density-0/1 dichotomies on uniform atoms, enabling a decomposition where the set is nearly a union of atoms of a high-rank quadratic factor. They prove significantly tighter bounds on the linear component of the regularity, achieving triple exponential growth for linear rank functions and quadruple exponential growth for higher-degree rank functions, thereby improving on previous tower-type bounds. The results parallel hypergraph regularity phenomena (as in Gishboliner–Shapira–Wigderson) and provide new tools for arithmetic regularity with potential applications in additive combinatorics and related areas.

Abstract

Previous work of the second author and Wolf showed that given a set $A\subseteq \mathbb{F}_p^n$ of bounded $\textrm{VC}_2$-dimension, there is a high rank quadratic factor $\mathcal{B}$ of bounded complexity such that $A$ is approximately equal to a union of atoms of $\mathcal{B}$. That proof yielded bounds of tower type on the linear and quadratic complexities. It was later shown by the same authors that the quadratic complexity can be improved to logarithmic, however that proof provided no improvement on the linear component. In this paper we prove that the bound on the linear complexity can be improved to a triple exponential in the case of linear rank functions, and a quadruple exponential for polynomial rank functions of higher degree. Our strategy is based on the one developed by Gishboliner, Wigderson, and Shapira to prove the analogous result in the hypergraph setting. Step 1 is to prove a``cylinder" version of the quadratic arithmetic regularity lemma, which says that given a set $A\subseteq G=\mathbb{F}_p^n$, there is a partition of $G$ into atoms of (possibly distinct) quadratic factors of high rank and bounded complexity, so that most atoms in the partition are uniform with respect to the set $A$, in the sense of a certain local $U^3$ norm. Step 2 is to show that if $A$ has bounded $\textrm{VC}_2$-dimension, then it has density near $0$ or $1$ on all atoms which are uniform in the sense of Step 1. Step 1 relies on a recent local version of the $U^3$ inverse theorem due to Prendiville, and is necessarily phrased in terms of a local $U^3$ norm implicit in that paper. On the other hand, Step 2 relies on a counting lemma for a different local $U^3$ due to Terry and Wolf, which we prove here is approximately the same as the local $U^3$ norm used in Step 1.

On the linear complexity of subsets of $\mathbb{F}_p^n$ bounded $\textrm{VC}_2$-dimension

TL;DR

This work advances our understanding of the structure of subsets of with bounded -dimension by introducing a cylinder-style quadratic arithmetic regularity lemma and a refined local -norm framework. The authors connect localized inverse theorems with density-0/1 dichotomies on uniform atoms, enabling a decomposition where the set is nearly a union of atoms of a high-rank quadratic factor. They prove significantly tighter bounds on the linear component of the regularity, achieving triple exponential growth for linear rank functions and quadruple exponential growth for higher-degree rank functions, thereby improving on previous tower-type bounds. The results parallel hypergraph regularity phenomena (as in Gishboliner–Shapira–Wigderson) and provide new tools for arithmetic regularity with potential applications in additive combinatorics and related areas.

Abstract

Previous work of the second author and Wolf showed that given a set of bounded -dimension, there is a high rank quadratic factor of bounded complexity such that is approximately equal to a union of atoms of . That proof yielded bounds of tower type on the linear and quadratic complexities. It was later shown by the same authors that the quadratic complexity can be improved to logarithmic, however that proof provided no improvement on the linear component. In this paper we prove that the bound on the linear complexity can be improved to a triple exponential in the case of linear rank functions, and a quadruple exponential for polynomial rank functions of higher degree. Our strategy is based on the one developed by Gishboliner, Wigderson, and Shapira to prove the analogous result in the hypergraph setting. Step 1 is to prove a``cylinder" version of the quadratic arithmetic regularity lemma, which says that given a set , there is a partition of into atoms of (possibly distinct) quadratic factors of high rank and bounded complexity, so that most atoms in the partition are uniform with respect to the set , in the sense of a certain local norm. Step 2 is to show that if has bounded -dimension, then it has density near or on all atoms which are uniform in the sense of Step 1. Step 1 relies on a recent local version of the inverse theorem due to Prendiville, and is necessarily phrased in terms of a local norm implicit in that paper. On the other hand, Step 2 relies on a counting lemma for a different local due to Terry and Wolf, which we prove here is approximately the same as the local norm used in Step 1.

Paper Structure

This paper contains 20 sections, 40 theorems, 163 equations.

Table of Contents

  1. Introduction
  2. Arithmetic regularity
  3. Hypergraph regularity
  4. Quadratic arithmetic regularity and main results
  5. Acknowledgements
  6. Notation
  7. Outline
  8. Preliminaries
  9. $\mathrm{VC}_2$-dimension
  10. Linear and quadratic factors
  11. Sizes of atoms and related sets
  12. Local $U^3$ norms and the local inverse theorem
  13. Prendiville's inverse theorem for the $U^3$-norm localized to a quadratic atom
  14. The local $U^3$-norms of Terry.2024d
  15. Relating the local $U^3$-norms
  16. ...and 5 more sections

Key Result

Theorem 1.1

For all odd primes $p$, all $\delta\in (0,1)$, and all growth functionsA growth function is an increasing function from $\mathbb{R}$ into $\mathbb{R}^{\geq 0}$.$\rho$, there is a constant $M(p,\delta,\rho)$ such that the following holds. For all $A\subseteq \mathbb{F}_p^n$, there are integers $\ell,

Theorems & Definitions (95)

  • Theorem 1.1: Proposition 4.2 in Terry.2021d
  • Theorem 1.2: Theorem 1.3 in Terry.2021f
  • Theorem 1.3
  • Theorem 1.4: "Cylinder" quadratic arithmetic regularity lemma
  • Definition 1.5: Growth functions
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Linear factors and atoms
  • Definition 2.4: Linear form associated to a linear factor
  • proof
  • ...and 85 more