Mixing for three-term progressions in finite simple groups

Sarah Peluse

doi:10.1017/S0305004117000482

Mixing for three-term progressions in finite simple groups

Sarah Peluse

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

Answering a question of Gowers, Tao proved that any A × B × C ⊂ SL_d(F_q)³ contains |A||B||C|/|SL_d(F_q)| + O_d(|SL_d(F_q)|²/q^min(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(F_q) 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(q)|²/q^(d-1)/24).

Original language	English (US)
Pages (from-to)	279-286
Number of pages	8
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Volume	165
Issue number	2
DOIs	https://doi.org/10.1017/S0305004117000482
State	Published - Sep 1 2018

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1017/S0305004117000482

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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(q)|2/q(d-1)/24).",

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AB - 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(q)|2/q(d-1)/24).

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Mixing for three-term progressions in finite simple groups

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