Szemeredi's regularity lemma is a fundamental result in graph theory, which says that sufficiently large finite graphs can be partitioned into a small number of pieces so that the edges between most pairs of

pieces are randomly distributed. In other words, the regularity lemma processes finite graphs into ingredients that are either highly structured (namely, the partition of the graph) or highly random (namely, the edges between regular pairs). In 2005, Green introduced arithmetic regularity, which is a group-theoretic analogue of Szemeredi's result. Green's work involves decompositions of finite abelian groups into algebraically structured pieces, which behave randomly with respect to some chosen subset

of the group.

In this talk, I will consider subsets $A$ of arbitrary groups $G$, such that the bipartite graph determined by $x\cdot y\in A$ omits some bipartite graph of a fixed finite size. I will present strengthened versions arithmetic regularity in this setting, which yield algebraic structure theorems for sets $A$ as above. This work combines tools from model theory, additive combinatorics, and the structure of compact topological groups.

Joint with A. Pillay and C. Terry.