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Mixing for progressions in non-abelian groups
We study the mixing properties of progressions ,
of length three and four in a model class of finite
non-abelian groups, namely the special linear groups over a finite
field , with bounded. For length three progressions , we
establish a strong mixing property (with error term that decays polynomially in
the order of ), which among other things counts the number of such
progressions in any given dense subset of , answering a question
of Gowers for this class of groups. For length four progressions
, we establish a partial result in the case if the
shift 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.Comment: 31 pages, no figures, to appear, Forum of Mathematics, Sigma. Referee
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