MIXING for PROGRESSIONS in NONABELIAN GROUPS

Abstract

We study the mixing properties of progressions (x, xg, xg2), (x, xg, xg2, xg3) of length three and four in a model class of finite nonabelian groups, namely the special linear groups SLd(F) over a finite field F, with d bounded. For length three progressions (x, xg, xg2), we establish a strong mixing property (with an 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 SLd (F), answering a question of Gowers for this class of groups. For length four progressions (x, xg, xg2, xg3), we establish a partial result in the d= 2 case if the shift g is restricted to be diagonalizable over F, 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 Szemerédi theorem.