Mixing for three-term progressions in finite simple groups

Abstract

AbstractAnswering a question of Gowers, Tao proved that any A × B × C ⊂ SLd(𝔽q)3 contains |A||B||C|/|SLd(𝔽q)| + Od(|SLd(𝔽q)|2/qmin(d−1,2)/8) three-term progressions (x, xy, xy2). 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 PSL2(𝔽q) 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(|SLd(𝔽q)|2/q(d−1)/24).