Product mixing in the alternating group

Sean Eberhard*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)

Abstract

We prove the following one-sided product-mixing theorem for the alternating group: Given subsets X, Y, Z ⊂ An of densities α,β, γ satisfying min(αβ,αγ,βγ)≫n-1(logn)7, there are at least (1+o(1))αβγ|An|2 solutions to xy = z with x ∈ X, y ∈ Y, z ∈ Z. One consequence is that the largest product-free subset of An has density at most n-1/2(logn)7/2, which is best possible up to logarithms and improves the best previous bound of n-1/3 due to Gowers. The main tools are a Fourier-analytic reduction noted by Ellis and Green to a problem just about the standard representation, a Brascamp-Lieb-type inequality for the symmetric group due to Carlen, Lieb, and Loss, and a concentration of measure result for rearrangements of inner products.

Original languageEnglish
Number of pages18
JournalDiscrete Analysis
Volume2
Issue number2016
DOIs
Publication statusPublished - 28 Feb 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2016 Sean Eberhard.

Keywords

  • Alternating group
  • Mixing in groups
  • Product-free sets
  • Symmetric group

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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