Title: Subspaces in difference sets in vector spaces
Speaker: Thai Hoang Le (University of Mississippi)
Abstract: A common theme in additive combinatorics states that if $A$ is a subset of positive density of a vector space $\mathbf{F}_p^n$, then the difference set $A-A$ must contain a large subspace. Furthermore, the more sums or differences we take (e.g. $A+A-A-A$), the larger subspaces we are guaranteed to find. I will talk about two results in this theme. In joint work with Zhenchao Ge, we generalize and simplify the proof of a result of Sanders, which says that if $A \subset \mathbf{F}_p^n$ has density $1/2-c(p)/\sqrt{n}$, then $A-A$ contains a subspace of codimension 1. In joint work with Pierre-Yves Bienvenu, we show that if $A$ is a subset of positive density of $\mathbf{F}_p^n \times \mathbf{F}_p^n$, then by doing more restrictive operations on the coordinates of $A$, the resulting set also contains nice structures. The same result was obtained independently by Gowers and Milicevic. We will also discuss a number theoretic application of this result, namely a function field instance of the Möbius randomness principle.
