Mixing of 3-Term Progressions in Quasirandom Groups

Authors Amey Bhangale, Prahladh Harsha , Sourya Roy

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Author Details

Amey Bhangale
  • University of California, Riverside, CA, USA
Prahladh Harsha
  • Tata Institute of Fundamental Research, Mumbai, India
Sourya Roy
  • University of California, Riverside, CA, USA


We thank Shachar Lovett and Terence Tao for helpful comments on an earlier draft of this paper. We are also thankful to the anonymous reviewers whose comments helped in improving the presentation of the paper.

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Amey Bhangale, Prahladh Harsha, and Sourya Roy. Mixing of 3-Term Progressions in Quasirandom Groups. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 20:1-20:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Quasirandom groups
  • 3-term arithmetic progressions


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