Group representations that resist worst-case sampling

Abstract

Motivated by expansion in Cayley graphs, we show that there exist infinitely many groups $G$ with a nontrivial irreducible unitary representation whose average over every set of $o(\log\log|G|)$ elements of $G$ has operator norm $1 - o(1)$. This answers a question of Lovett, Moore, and Russell, and strengthens their negative answer to a question of Wigderson. The construction is the affine group of $\mathbb{F}_p$ and uses the fact that for every $A \subset \mathbb{F}_p\setminus\{0\}$, there is a set of size $\exp(\exp(O(|A|)))$ that is almost invariant under both additive and multiplicatpive translations by elements of $A$.