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- W2123754135 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." @default.
- W2123754135 created "2016-06-24" @default.
- W2123754135 creator A5066728260 @default.
- W2123754135 date "2012-12-11" @default.
- W2123754135 modified "2023-10-16" @default.
- W2123754135 title "Mixing for progressions in non-abelian groups" @default.
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- W2123754135 doi "https://doi.org/10.48550/arxiv.1212.2586" @default.
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