### A stable arithmetic regularity lemma in finite abelian groups

Analysis and Geometry Seminar

29th May 2018, 3:00 pm – 4:00 pm

Howard House, 2nd Floor Seminar Room

The arithmetic regularity lemma for *F _{p}^{n}* (first proved by Green in 2005) states that given

*A ⊆ F*, there exists

_{p}^{n}*H ≤ F*of bounded index such that

_{p}^{n}*A*is Fourier-uniform with respect to almost all cosets of

*H*. In general, the growth of the index of

*H*is required to be of tower type depending on the degree of uniformity, and must also allow for a small number of non-uniform elements. Previously, in joint work with Wolf, we showed that under a natural stability theoretic assumption, the bad bounds and non-uniform elements are not necessary. In this talk, we present results extending these results to stable subsets of arbitrary finite abelian groups. This is joint work with Julia Wolf.

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