Efficient Arithmetic Regularity and Removal Lemmas for Induced Bipartite Patterns

Let G be an abelian group of bounded exponent and A subset of 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 vertical bar G vertical bar 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.