Mixing for three-term progressions in finite simple groups

Abstract

Answering a question of Gowers, Tao proved that any $A\times B\times C\subset S{L}_d({\mathbb{F}}_q{)}^3$ contains $|A||B||C|/|S{L}_d({\mathbb{F}}_q)|+O_d(|S{L}_d({\mathbb{F}}_q){|}^2/q^{\min (d-1,2)/8})$ three-term progressions $(x,xy,x{y}^2)$. 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 $O(|S{L}_d({\mathbb{F}}_q){|}^2/q^{(d-1)/24})$.

10 pages; v2: fixed a typo

Cited by in corpus (2)

• Complete type amalgamation for non-standard finite groups
• Mixing of 3-term progressions in Quasirandom Groups