##### Speaker

Dr. Thai Haong Le, University of Mississippi

##### Title

Structures in difference sets

##### Physical Location

Allen Hall 411

**Abstract:** A general principle says that if *A* is a subset of a group *G* and *A* is large in a certain sense, then we expect the difference set *A - A* = {*a* - *a*’: *a*, *a*’ ∈ *A* } to contain nice structures. For example, a classical theorem of Steinhaus says that if *A* is a subset of positive Lebesgue measure of ℝ, then *A - A* contains an open interval around 0. If *A* is a subset of positive density of a vector space ?_{p}^{n} over a finite field ?_{p}, then we expect *A - A* to contain a large subspace. I will talk about various manifestations of this principle in analysis, number theory and combinatorics. These include joint works with P.-Y. Bienvenu and Z. Ge.