Gabriel Conant

VC-dimension, pseudofinite groups, and arithmetic regularity

I will discuss recent work on the structure of VC-sets in groups, i.e. subsets whose family of left translates has absolutely bounded VC-dimension. In joint work with A. Pillay, we show that VC-set A in a pseudofinite group G is "generically dominated" by a certain compact group of the form G/H, where H is a canonical normal subgroup of G associated to A. Informally, this implies that almost all cosets of H are either almost contained in A or almost disjoint from A. More formally, if C is the set of cosets of H, which intersect both A and its complement in large sets with respect to the pseudofinite measure on G, then the Haar measure of C in G/H is zero.

In joint work with A. Pillay and C. Terry, we use this generic domination to prove arithmetic regularity lemmas for VC-sets in finite groups. These results are motivated by Green's arithmetic regularity lemma for abelian groups, and generalize (without effective bounds), work of Alon-Fox-Zhao and Terry-Wolf on improved arithmetic regularity for VC-sets.

Gabriel Conant is a Lumpkins Postdoctoral Fellow at the University of Notre Dame. He studies the Model theory of groups and homogeneous structures. This semester he is at the Institut Henri Poincare attending the Model Theory, Combinatorics and Valued Fields trimester program.