Erratum: Mixing, Communication Complexity and Conjectures of Gowers and Viola (Combinatorics, Probability and Computing (2017) 26 (628-640) DOI: 10.1017/S096354831600016X)

Aner Shalev*

*Corresponding author for this work

Research output: Contribution to journalComment/debate

1 Scopus citations

Abstract

In my paper [1], a normalization factor of |G|-1was missing in the statements of Theorem 2.4, Theorem 2.6 and Corollary 2.7. The correct formulation of these results is as follows. Theorem 2.4. Let G be a finite simple group of Lie type of rank r over a field with q elements. (i) There is a constant c > 0 depending only on r, such that, if x, y are distributed uniformly over G (but may be dependent), then 'Equation Presented' holds with probability at least 1 - |G|-c. (ii) There is an absolute constant c > 0 and a constant c′ depending only on r, such that, if G ∉ S, where 'Equation Presented' and x, y are distributed uniformly over G (but may be dependent), then 'Equation Presented' holds with probability at least 1 - c/q. Theorem 2.6. Let G be a finite simple group. Let x, y be distributed uniformly over G (but they may be dependent). Fix s with s > 0. (i) If G = An, then for some absolute constant c the probability that 'Equation Presented' is at least 1 - cn-s. (ii) If G is a finite simple group of Lie type of rank r over the field with q elements, then the probability that 'Equation Presented' is at least 1 - q-(s-ϵ)r, for any ϵ > 0 and r ≥ r(s, ϵ). Corollary 2.7. Let G be a finite simple group. Let x, y be distributed uniformly over G (but they may be dependent). (i) If G = An, then for any ϵ > 0 there exists n(ϵ) such that, for any n ≥ n(ϵ), the probability that 'Equation Presented' is at least 1 - n-ϵ/3. (ii) If G is a finite simple group of Lie type of rank r over the field with q elements, then the probability that 'Equation Presented' is at least 1 - q-ϵr/3, for any ϵ > 0 and r ≥ r(ϵ). Consequently, the quotation of part (ii) above in the Introduction should change accordingly. No changes whatsoever are required in the proofs.

Original languageAmerican English
Pages (from-to)954-955
Number of pages2
JournalCombinatorics Probability and Computing
Volume26
Issue number6
DOIs
StatePublished - 2017

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