Mixing of 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.