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August 2022 Locally Random Groups
Keivan Mallahi-Karai, Amir Mohammadi, Alireza Salehi Golsefidy
Michigan Math. J. 72: 479-527 (August 2022). DOI: 10.1307/mmj/20217213

Abstract

In this work, we introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls, and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we develop a Littlewood–Paley decomposition and explore its connection to the existence of a spectral gap for random walks. Moreover, under the dimension condition alone, we prove a multi-scale entropy gain result à la Bourgain–Gamburd and Tao.

Dedication

To Gopal Prasad on the occasion of his 75th birthday

Citation

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Keivan Mallahi-Karai. Amir Mohammadi. Alireza Salehi Golsefidy. "Locally Random Groups." Michigan Math. J. 72 479 - 527, August 2022. https://doi.org/10.1307/mmj/20217213

Information

Received: 5 May 2021; Revised: 3 January 2022; Published: August 2022
First available in Project Euclid: 2 August 2022

Digital Object Identifier: 10.1307/mmj/20217213

Subjects:
Primary: 22C05 , 43A77 , 60B15

Rights: Copyright © 2022 The University of Michigan

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Vol.72 • August 2022
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