Special Colloquium
Caroline Terry
University of Chicago
On the structure of stable subsets of finite abelian groups
Abstract: The arithmetic regularity lemma for $\mathbb{F}_p^n$ (first proved by Green in 2005) states that given a subset $A$ of $\mathbb{F}_p^n$, there exists a subspace $H\leq \mathbb{F}_p^n$, of bounded codimension, such that $A$ is Fourier-uniform with respect to almost all translates of $H$. This arithmetic regularity lemma can be seen as group theoretic analogue of Szermedi's graph regularity lemma, and has many generalizations and applications in arithmetic combinatorics. In general, the growth of the codimension 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. In this talk, we present joint work with Wolf, in which we show that under a natural model theoretic assumption, much stronger structural decomposition theorems can be proved. In particular, when the set in question is stable, we show that the bad bounds and non-uniformity are not necessary.
Wednesday February 5, 2020 at 3:00 PM in 636 SEO