2026-08-05T07:20:08Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15616 2024-03-06T10:55:46Z ddc:004 open_access

Mixing of 3-Term Progressions in Quasirandom Groups Bhangale, Amey Harsha, Prahladh Roy, Sourya Quasirandom groups 3-term arithmetic progressions 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Amey Bhangale and Prahladh Harsha and Sourya Roy 2022 Is Part Of LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ITCS.2022.20 urn:nbn:de:0030-drops-156163 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.20 eng https://creativecommons.org/licenses/by/4.0/legalcode