Quasirandom groups enjoy interleaved mixing

TL;DR

This paper demonstrates that quasirandom groups exhibit strong mixing properties when interleaving independent distributions, with bounds depending on the group's representation dimension.

Contribution

It generalizes, improves, and simplifies previous results on mixing properties of quasirandom groups with interleaved distributions.

Findings

Bound on the deviation of the product distribution from uniform

Dependence of mixing quality on the group's minimal representation dimension

Simplified proof of interleaved mixing properties

Abstract

Let $G$ be a group such that any non-trivial representation has dimension at least $d$. Let $X=(X_{1},X_{2},\ldots,X_{t})$ and $Y=(Y_{1},Y_{2},\ldots,Y_{t})$ be distributions over $G^{t}$. Suppose that $X$ is independent from $Y$. We show that for any $g\in G$ we have $|\mathbb{P}[X_{1}Y_{1}X_{2}Y_{2}\cdots X_{t}Y_{t}=g]-1/|G||\le\frac{|G|^{2t-1}}{d^{t-1}}\sqrt{\mathbb{E}_{h\in G^{t}}X(h)^{2}}\sqrt{\mathbb{E}_{h\in G^{t}}Y(h)^{2}}.$ Our results generalize, improve, and simplify previous works.