Mixing of 3-term progressions in Quasirandom Groups

TL;DR

This paper proves that three-term progressions in finite quasirandom groups exhibit mixing behavior, confirming a conjecture of Gowers, with an elementary proof based on basic nonabelian Fourier analysis.

Contribution

It provides a complete and elementary proof of Gowers' conjecture on 3-term progressions in quasirandom groups, extending previous results to all such groups.

Findings

Establishes mixing of 3-term progressions in all finite quasirandom groups.

Provides an elementary proof using basic nonabelian Fourier analysis.

Confirms Gowers' conjecture for 3-term progressions in quasirandom groups.

Abstract

In this note, we show the mixing of three-term progressions $(x, xg, xg^2)$ in every finite quasirandom groups, fully answering a question of Gowers. More precisely, we show that for any $D$-quasirandom group $G$ and any three sets $A_1, A_2, A_3 \subset G$, we have \[ \left|\Pr_{x,y\sim G}\left[ x \in A_1, xy \in A_2, xy^2 \in A_3\right] - \prod_{i=1}^3 \Pr_{x\sim G}\left[x \in A_i\right] \right| \leq \left(\frac{2}{\sqrt{D}}\right)^{\frac{1}{4}}.\] Prior to this, Tao answered this question when the underlying quasirandom group is $\mathrm{SL}_{d}(\mathbb{F}_q)$. Subsequently, Peluse extended the result to all nonabelian finite $\textit{simple}$ groups. In this work, we show that a slight modification of Peluse's argument is sufficient to fully resolve Gower's quasirandom conjecture for 3-term progressions. Surprisingly, unlike the proofs of Tao and Peluse, our proof is elementary and only uses basic facts from nonabelian Fourier analysis.