## Arithmetic Regularity Lemmas

Szemeredi's regularity lemma is an indispensible tool in graph theory. It is often described as giving a "structure theorem for all graphs". I will describe a result, or rather a hierarchy of results, which does the same with "subsets of abelian groups" in place of "graphs". I will talk about applications of this result, such as (i) If a subset of {1,...,N} is almost sum-free (meaning that it has few solutions to x + y = z) then it may be expressed as a union of a genuinely sum-free set together with a small set; (ii) (Bergelson-Host-Kra conjecture) Fix alpha, epsilon. Let A be a subset of {1,...,N} with density alpha, where N is sufficiently large as a function of alpha and epsilon. Then there is some d such that A contains at least (alpha^3 - epsilon) N three-term progressions with common difference d, and some d' such that A contains at least (alpha^4 - epsilon) N four-term progressions with common difference d'. The analogue of this result fails for progressions of length five and longer. Some of this work is joint with T. Tao.

**Ben Green, Clay Institute, University of Bristol, and M.I.T. **