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MIXING FOR PROGRESSIONS IN NONABELIAN GROUPS

Part of: Representation theory of groups Sequences and sets

Published online by Cambridge University Press:  29 August 2013

TERENCE TAO
TERENCE TAO*
Affiliation:
UCLA Department of Mathematics, Los Angeles, CA 90095-1555, USAtao@math.ucla.edu

Abstract

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We study the mixing properties of progressions $(x, xg, x{g}^{2} )$, $(x, xg, x{g}^{2} , x{g}^{3} )$ of length three and four in a model class of finite nonabelian groups, namely the special linear groups ${\mathrm{SL} }_{d} (F)$ over a finite field $F$, with $d$ bounded. For length three progressions $(x, xg, x{g}^{2} )$, we establish a strong mixing property (with an error term that decays polynomially in the order $\vert F\vert $ of $F$), which among other things counts the number of such progressions in any given dense subset $A$ of ${\mathrm{SL} }_{d} (F)$, answering a question of Gowers for this class of groups. For length four progressions $(x, xg, x{g}^{2} , x{g}^{3} )$, 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.

MSC classification

Secondary: 11B30: Arithmetic combinatorics; higher degree uniformity 20D60: Arithmetic and combinatorial problems
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 1 , 2013 , e2
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2013.

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