Mixing, Communication Complexity and Conjectures of Gowers and Viola

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Abstract

We study the distribution of products of conjugacy classes in finite simple groups, obtaining effective two-step mixing results, which give rise to an approximation to a conjecture of Thompson. Our results, combined with work of Gowers and Viola, also lead to the solution of recent conjectures they posed on interleaved products and related complexity lower bounds, extending their work on the groups SL(2, q) to all (non-abelian) finite simple groups. In particular it follows that, if G is a finite simple group, and A, B ⊂ Gt for t ≥ 2 are subsets of fixed positive densities, then, as a = (a1,..., at ) ∈ A and b = (b 1,..., bt ) ∈ B are chosen uniformly, the interleaved product a • b:=a1b1.. atbt is almost uniform on G (with quantitative estimates) with respect to the ℓ-norm. It also follows that the communication complexity of an old decision problem related to interleaved products of a, b ∈ Gt is at least Ω(t log |G|) when G is a finite simple group of Lie type of bounded rank, and at least Ω(t log log |G|) when G is any finite simple group. Both these bounds are best possible.

Original languageAmerican English
Pages (from-to)628-640
Number of pages13
JournalCombinatorics Probability and Computing
Volume26
Issue number4
DOIs
StatePublished - 1 Jul 2017

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Statistics and Probability
  • Computational Theory and Mathematics
  • Applied Mathematics

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