License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.20
URN: urn:nbn:de:0030-drops-156163
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15616/
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Bhangale, Amey ; Harsha, Prahladh ; Roy, Sourya

Mixing of 3-Term Progressions in Quasirandom Groups

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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}(?_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.

BibTeX - Entry

@InProceedings{bhangale_et_al:LIPIcs.ITCS.2022.20,
  author =	{Bhangale, Amey and Harsha, Prahladh and Roy, Sourya},
  title =	{{Mixing of 3-Term Progressions in Quasirandom Groups}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{20:1--20:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15616},
  URN =		{urn:nbn:de:0030-drops-156163},
  doi =		{10.4230/LIPIcs.ITCS.2022.20},
  annote =	{Keywords: Quasirandom groups, 3-term arithmetic progressions}
}

Keywords: Quasirandom groups, 3-term arithmetic progressions
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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