Mixing for progressions in non-abelian groups

TL;DR

This paper investigates the mixing properties of specific progressions in finite non-abelian groups, particularly $SL_d(F)$, providing new bounds and partial results for progressions of length three and four.

Contribution

It establishes strong mixing results for length three progressions in $SL_d(F)$ and offers partial results for length four progressions with diagonalizable shifts, answering Gowers' question for this class.

Findings

Polynomial decay error bounds for length three progressions

Counting results for progressions in dense subsets of $SL_d(F)$

Partial results for length four progressions with diagonalizable shifts

Abstract

We study the mixing properties of progressions $(x,xg,xg^2)$, $(x,xg,xg^2,xg^3)$ of length three and four in a model class of finite non-abelian groups, namely the special linear groups $SL_d(F)$ over a finite field $F$, with $d$ bounded. For length three progressions $(x,xg,xg^2)$, we establish a strong mixing property (with error term that decays polynomially in the order $|F|$ of $F$), which among other things counts the number of such progressions in any given dense subset $A$ of $SL_d(F)$, answering a question of Gowers for this class of groups. For length four progressions $(x,xg,xg^2,xg^3)$, we establish a partial result in the $d=2$ case if the shift $g$ is restricted to be diagonalisable over the field, although in this case we do not recover polynomial bounds in the error term. Our methods include the use of the Cauchy-Schwarz inequality, the abelian Fourier transform, the Lang-Weil bound for the number of points in an algebraic variety over a finite field, some algebraic geometry, and (in the case of length four progressions) the multidimensional Szemer\'edi theorem.