Mixing of 3-term progressions in Quasirandom Groups

Amey Bhangale; Prahladh Harsha; Sourya Roy

doi:10.4230/LIPIcs.ITCS.2022.20

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Mixing of 3-term progressions in Quasirandom Groups

Conference proceeding

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Mixing of 3-term progressions in Quasirandom Groups

Amey Bhangale, Prahladh Harsha and Sourya Roy

Leibniz International Proceedings in Informatics, LIPIcs, Vol.215

01/25/2022

DOI: 10.4230/LIPIcs.ITCS.2022.20

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Abstract

In this paper, we show the mixing of three-term progressions (x, xg, xg²) in every finite quasirandom group, fully answering a question of Gowers. More precisely, we show that for any D-quasirandom group G and any three sets A₁, A₂, A₃ ⊂ G, we have |Pr_{x,y∼ G}[x ∈ A₁, xy ∈ A₂, xy² ∈ A₃] - ∏_{i = 1}³ Pr_{x∼ G}[x ∈ A_i]| ≤ (2/(√{D)})^{1/4}. Prior to this, Tao answered this question when the underlying quasirandom group is SL_{d}(𝔽_q). Subsequently, Peluse extended the result to all non-abelian finite simple groups. In this work, we show that a slight modification of Peluse’s argument is sufficient to fully resolve Gowers' quasirandom conjecture for 3-term progressions. Surprisingly, unlike the proofs of Tao and Peluse, our proof is elementary and only uses basic facts from non-abelian Fourier analysis.

Quasirandom groups

3-term arithmetic progressions

Details

Title: Subtitle: Mixing of 3-term progressions in Quasirandom Groups
Creators: Amey Bhangale
Prahladh Harsha
Sourya Roy
Resource Type: Conference proceeding
Publication Details: Leibniz International Proceedings in Informatics, LIPIcs, Vol.215
DOI: 10.4230/LIPIcs.ITCS.2022.20
ISSN: 1868-8969
eISSN: 2331-8422
Language: English
Date published: 01/25/2022
Academic Unit: Computer Science
Record Identifier: 9984446404602771

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