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On a conjecture of Terry and Wolf

V. Gladkova

Abstract

This paper shows that the $\mathrm{VC}_2$-dimension of a subset of $\mathbb{F}_p^n$ known as the 'quadratic Green-Sanders example' is at least 3 and at most 501. The upper bound confirms a conjecture of Terry and Wolf, who introduced this set in their recent work concerning strengthenings of the higher-order arithmetic regularity lemma under certain model-theoretic tameness assumptions. Additionally, the paper presents a simplified proof that the (linear) Green-Sanders example, which has its roots in Ramsey theory, has $\mathrm{VC}$-dimension at most 3.

On a conjecture of Terry and Wolf

Abstract

This paper shows that the -dimension of a subset of known as the 'quadratic Green-Sanders example' is at least 3 and at most 501. The upper bound confirms a conjecture of Terry and Wolf, who introduced this set in their recent work concerning strengthenings of the higher-order arithmetic regularity lemma under certain model-theoretic tameness assumptions. Additionally, the paper presents a simplified proof that the (linear) Green-Sanders example, which has its roots in Ramsey theory, has -dimension at most 3.

Paper Structure

This paper contains 5 sections, 11 theorems, 31 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. The $\mathrm{VC}$-dimension of the linear Green-Sanders example
  3. The quadratic Green-Sanders example has bounded $\mathrm{VC}_2$-dimension
  4. A lower bound on the $\mathrm{VC}_2$-dimension of the quadratic Green-Sanders example
  5. Multicolour bipartite Ramsey numbers

Key Result

Theorem 1.3

Fix $k \geq 1$ and $\epsilon > 0$. Suppose that $S \subseteq \mathbb{F}_{p}^{n}$ has $\mathrm{VC}$-dimension of at most $k$. Then there is a subspace $H \leqslant \mathbb{F}_{p}^{n}$ of index at most $\epsilon^{-k-o(1)}$ such that for all but an $\epsilon$-proportion of $c \in \mathbb{F}_{p}^{n}$, e

Figures (1)

  • Figure 1: Green-Sanders examples in $\mathbb{F}_{3}^{n}$. $\mathrm{GS}(3, n)$ and $\mathrm{QGS}(3, n)$, defined by level sets of linear and quadratic polynomials respectively.

Theorems & Definitions (29)

  • Definition 1.1: Linear Green-Sanders example
  • Definition 1.2: $\mathrm{VC}$-dimension of a subset
  • Theorem 1.3: Arithmetic regularity lemma for sets of bounded $\mathrm{VC}$-dimension bounded-vc-arlsisask
  • Theorem 1.4
  • Definition 1.5: $\mathrm{VC}_2$-dimension of a subset
  • Definition 1.6: Quadratic Green-Sanders example
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1: $\mathrm{VC}$-dimension of a family of sets
  • Theorem 2.2
  • ...and 19 more