Submission date: 12 February 2018

Abstract:

Suppose G is a finite group and A ⊆ G is such that {gA : g in G} has VC-dimension strictly less than k. We find algebraically well-structured sets in G which, up to a chosen ε > 0, describe the structure of A and behave regularly with respect to translates of A. For the subclass of groups with uniformly fixed finite exponent r, this algebraic object is a subgroup whose index is bounded in terms of k, r, and ε. For arbitrary groups, we use Bohr neighborhoods inside of subgroups of bounded index. Both results are proved using model theoretic techniques for NIP formulas in pseudofinite groups.

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Full text arXiv 1802.04246: pdf, ps.