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Mixing for three-term progressions in finite simple groups

Published online by Cambridge University Press:  25 May 2017

SARAH PELUSE
SARAH PELUSE*
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Bldg. 380, Stanford, CA 94305, U.S.A. e-mail: speluse@stanford.edu
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Abstract

Answering a question of Gowers, Tao proved that any A × B × C ⊂ SLd(𝔽q)3 contains |A||B||C|/|SLd(𝔽q)| + Od(|SLd(𝔽q)|2/qmin(d−1,2)/8) three-term progressions (x, xy, xy2). Using a modification of Tao's argument, we prove such a mixing result for three-term progressions in all nonabelian finite simple groups except for PSL2(𝔽q) with an error term that depends on the degree of quasirandomness of the group. This argument also gives an alternative proof of Tao's result when d > 2, but with the error term O(|SLd(𝔽q)|2/q(d−1)/24).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 2 , September 2018 , pp. 279 - 286
Copyright
Copyright © Cambridge Philosophical Society 2017 

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Footnotes

†

Supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-114747 and by the Stanford University Mayfield Graduate Fellowship.

References

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