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15 February 2020 On the width of transitive sets: Bounds on matrix coefficients of finite groups
Ben Green
Duke Math. J. 169(3): 551-578 (15 February 2020). DOI: 10.1215/00127094-2019-0074

Abstract

We say that a finite subset of the unit sphere in Rd is transitive if there is a group of isometries which acts transitively on it. We show that the width of any transitive set is bounded above by a constant times (logd)1/2.

This is a consequence of the following result: if G is a finite group and ρ:GUd(C) a unitary representation, and if vCd is a unit vector, then there is another unit vector wCd such that sup gG|ρ(g)v,w|(1+clogd)1/2.

These results answer a question of Yufei Zhao. An immediate consequence of our result is that the diameter of any quotient S(Rd)/G of the unit sphere by a finite group G of isometries is at least π/2od(1).

Citation

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Ben Green. "On the width of transitive sets: Bounds on matrix coefficients of finite groups." Duke Math. J. 169 (3) 551 - 578, 15 February 2020. https://doi.org/10.1215/00127094-2019-0074

Information

Received: 6 March 2018; Revised: 4 October 2019; Published: 15 February 2020
First available in Project Euclid: 9 January 2020

zbMATH: 07198461
MathSciNet: MR4065149
Digital Object Identifier: 10.1215/00127094-2019-0074

Subjects:
Primary: 20D06
Secondary: 51F25

Rights: Copyright © 2020 Duke University Press

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Vol.169 • No. 3 • 15 February 2020
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