Efficient arithmetic regularity and removal lemmas for induced bipartite patterns

TL;DR

This paper proves that sets with low VC dimension in abelian groups can be approximated by unions of subgroup cosets, and establishes a polynomial bounds removal lemma for induced bipartite patterns, advancing property testing methods.

Contribution

It introduces a new regularity lemma for sets with bounded VC dimension and a polynomial bounds removal lemma for induced bipartite patterns in finite abelian groups.

Findings

Sets with low VC dimension can be approximated by unions of subgroup cosets.

A polynomial bounds removal lemma for induced bipartite patterns is established.

Applications to property testing in finite abelian groups are demonstrated.

Abstract

Let $G$ be an abelian group of bounded exponent and $A \subseteq G$. We show that if the collection of translates of $A$ has VC dimension at most $d$, then for every $\epsilon>0$ there is a subgroup $H$ of $G$ of index at most $\epsilon^{-d-o(1)}$ such that one can add or delete at most $\epsilon|G|$ elements to/from $A$ to make it a union of $H$-cosets. We also establish a removal lemma with polynomial bounds, with applications to property testing, for induced bipartite patterns in a finite abelian group with bounded exponent.