Mixing for three-term progressions in finite simple groups

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-termprogressions$(x,xy,xy^2)$. 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 $PS{L}_2({\mathbb{F}}_q)$ 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 $O(\mathrm{\mid }S{L}_d({\mathbb{F}}_q){\mathrm{\mid }}^2\mathrm{/}{q}^{(d-1)\mathrm{/}24})$.