Convolutions of sets with bounded VC-dimension are uniformly continuous

TL;DR

This paper introduces a VC-dimension concept for subsets of groups, showing that bounded VC-dimension implies strong uniform continuity of convolution, with implications for the Polynomial Bogolyubov--Ruzsa Conjecture and structure theorems.

Contribution

It generalizes the stable arithmetic regularity lemma and proves the Polynomial Bogolyubov--Ruzsa Conjecture for sets with bounded VC-dimension.

Findings

Bounded VC-dimension implies Bohr uniform continuity of convolution.

Supports the Polynomial Bogolyubov--Ruzsa Conjecture for VC-bounded sets.

Provides a structure theorem for translation-closed VC-bounded set systems.

Abstract

We study a notion of VC-dimension for subsets of groups, defining this for a set $A$ to be the VC-dimension of the family $\{ (xA) \cap A : x \in A\cdot A^{-1} \}$. We show that if a finite subset $A$ of an abelian group has bounded VC-dimension, then the convolution $1_A*1_{-A}$ is Bohr uniformly continuous, in a quantitatively strong sense. This generalises and strengthens a version of the stable arithmetic regularity lemma of Terry and Wolf in various ways. In particular, it directly implies that the Polynomial Bogolyubov--Ruzsa Conjecture -- a strong version of the Polynomial Freiman--Ruzsa Conjecture -- holds for sets with bounded VC-dimension. We also prove some results in the non-abelian setting. In some sense, this gives a structure theorem for translation-closed set systems with bounded (classical) VC-dimension: if a VC-bounded family of subsets of an abelian group is closed under translation, then each member has a simple description in terms of Bohr sets, up to a small error.