Abstract

We say a subset S of a group G is product-free if there are no solutions to the equation xy=z with x,y and z all elements of S. We make progress on a question of Gowers concerning the maximal measure of a measurable product-free subset of SU(n), showing that this measure is at most exp(-cn^{1/3}) for all n, where c>0 is an absolute constant. In fact, we prove similar ‘stretched exponential’ upper bounds for any compact, connected Lie group, in terms of the minimal dimension of a nontrivial irreducible continuous ordinary representation of the group. The best-known lower bound (for SU(n)) remains exp(-cn) for an absolute constant c, which may well be the truth. Our bounds for SU(n) are best-possible for a slightly stronger phenomenon, known as product-mixing.

Our main tools are some new hypercontractive inequalities, one of which says that if f is a real-valued function on SU(n), then the q-norm of Tf (for some quite large value of q > 2) is no larger than the 2-norm of f, where T is a natural `noise’ operator which, roughly speaking, replaces the value of f at x with the average of f over a small sphere centred at x (for each x). Our methods are easiest to explain if we replace SU(n) with SO(n), where we obtain the desired hypercontractive inequality by a coupling with Gaussian space, combined with a variant of Weyl’s unitary trick. In this talk I will try to give a good (high-level) picture of these methods.