Mixing for three-term progressions in finite simple groups

Abstract

Answering a question of Gowers, Tao proved that any $A\times B\times C\subset SL_d(\mathbb{F}_q)^3$ contains $|A||B||C|/|SL_d(\mathbb{F}_q)|+O_d(|SL_d(\mathbb{F}_q)|^2/q^{\min(d-1,2)/8})$ three-term progressions $(x,xy,xy^2)$. Using a modification of Tao's argument, we prove such a mixing result for three-term progressions in all nonabelian finite simple groups except for $PSL_2(\mathbb{F}_q)$ with an error term that depends on the degree of quasirandomness of the group. This argument also gives an alternative proof of Tao's result when $d>2$, but with the error term $O(|SL_d(\mathbb{F}_q)|^2/q^{(d-1)/24})$.