A Stable Arithmetic Regularity Lemma in Finite Abelian Groups

The arithmetic regularity lemma for $\mathbb{F}_p^n$ (first proved by Green in 2005) states that given $A\subseteq \F_p^n$, there exists $H\leq \F_p^n$ of bounded index such that $A$ 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.