Convolutions of sets with bounded VC-dimension are uniformly continuous

Olof Sisask

Published 2018-02-08Version 1

We introduce a notion of VC-dimension for subsets of groups, defining this for a set $A$ to be the VC-dimension of the family $\{ A \cap(xA) : x \in A\cdot A^{-1} \}$. 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.