Isaac Goldbring : Syndeticity in amenable groups

Posted by Sherwood Hachtman , part of the Logic Seminar.

At

Jan. 19, 2016, 4 p.m.

In

SEO 427

Abstract

In the early 2000s, Renling Jin used nonstandard analysis to prove the following result: if $A$ and $B$ are subsets of the integers of positive Banach density, then $A+B$ is piecewise syndetic, meaning that there is a natural number $m$ such that $A+B+[-m,m]$ contains arbitrarily long intervals. Jin’s result had subsequently been improved upon in two different ways. First, Beiglbock, Bergelson, and Fish generalized Jin’s result to arbitrary amenable groups. Secondly, in joint work with DiNasso, Jin, Leth, Lupini, and Mahlburg, we proved a ``quantitative version'' of Jin’s result for the integers by giving a lower bound on the density of the set of witnesses to piecewise syndeticity. In this talk, I will prove a common generalization of these two results (again joint with DJLLM) by proving the amenable group version of our quantitative result.