# Group representations that resist worst-case sampling

## TL;DR

This paper constructs groups with irreducible representations that maintain high operator norm over small subsets, answering longstanding questions about group expansion and sampling resistance.

## Contribution

It demonstrates the existence of infinite groups with representations resistant to worst-case sampling, using affine groups over finite fields and novel almost-invariant set constructions.

## Key findings

- Existence of groups with representations resistant to small-sample averaging
- Construction based on affine groups over finite fields
- Sets that are almost invariant under additive and multiplicative translations

## 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 $A$.