Abstract

A common theme in additive combinatorics is that if a subset of a group is “approximately structured”, then it can be approximated by a set that is “perfectly structured”. In this talk, I will consider subsets A of groups G that are approximately structured in the sense that the bipartite graph defined by the relation “xy is in A” omits some bipartite graph, of a fixed finite size, as an induced subgraph. This can also be quantified using the VC-dimension of the set system of (left) translates of A. I will present several results showing that if a subset of an arbitrary finite group is approximately structured in this way, then it can be approximated by “perfectly structured” sets such as subgroups and Bohr sets. These results qualitatively generalize work of Terry and Wolf, and of Alon, Fox, and Zhao, on tame forms of arithmetic regularity in finite abelian groups. The proofs rely on model theory, as well as classical results from the structure theory for compact groups. Joint with A. Pillay and C. Terry.