Date:
Thu, 07/03/202410:30-11:30
Location:
Sprinzak 29
A subset A of SU(n) is said to be product free if for all a,b\in A ab is not in A. In a rather influential paper from 2006 Gowers conjectured that the largest product free subset of SU(n) has measure \le e^{-cn} and was able to obtain a polynomial bound of n^{-1/3}. We make a significant step towards Gowers conjecture by giving a bound of e^{-n^{1/3}} on the Haar measure of product free sets.
In order to do that we introduce tools from the Boolean cube (level-d inequalities) to the study of mixing and growth in compact Lie groups. We then show that these synergize perfectly with their representation theory to obtain our result.
Our methods also allow us to obtain the following rather counterintuitive result, which was conjectured by physicists independently of Gowers:
There exists c>0, such that if A is a subset of SU(n) of measure \ge e^{-c\sqrt{n}}, then the product of two Haar-random elements of A is equidistributed in SU(n).
Based on a joint work with Ellis, Kindler, and Minzer.