TY - JOUR
T1 - Erratum
T2 - Mixing, Communication Complexity and Conjectures of Gowers and Viola (Combinatorics, Probability and Computing (2017) 26 (628-640) DOI: 10.1017/S096354831600016X)
AU - Shalev, Aner
N1 - Publisher Copyright:
© 2017 Cambridge University Press. All rights reserved.
PY - 2017
Y1 - 2017
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85020612855&partnerID=8YFLogxK
U2 - 10.1017/S0963548317000414
DO - 10.1017/S0963548317000414
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AN - SCOPUS:85020612855
SN - 0963-5483
VL - 26
SP - 954
EP - 955
JO - Combinatorics Probability and Computing
JF - Combinatorics Probability and Computing
IS - 6
ER -