15 March 2021 Kleinian Schottky groups, Patterson–Sullivan measures, and Fourier decay
Jialun Li, Frédéric Naud, Wenyu Pan
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Duke Math. J. 170(4): 775-825 (15 March 2021). DOI: 10.1215/00127094-2020-0058


Let Γ be a Zariski-dense Kleinian Schottky subgroup of PSL2(C). Let ΛΓC be its limit set, endowed with a Patterson–Sullivan measure μ supported on ΛΓ. We show that the Fourier transform μˆ(ξ) enjoys polynomial decay as |ξ| goes to infinity. As a corollary, all limit sets of Zariski-dense Kleinian groups have positive Fourier dimension. This is a PSL2(C) version of the PSL2(R) result of Bourgain and Dyatlov, and uses the decay of exponential sums based on Bourgain–Gamburd’s sum-product estimate on C. These bounds on exponential sums require a delicate nonconcentration hypothesis which is proved using some representation theory and regularity estimates for stationary measures of certain random walks on linear groups.


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Jialun Li. Frédéric Naud. Wenyu Pan. "Kleinian Schottky groups, Patterson–Sullivan measures, and Fourier decay." Duke Math. J. 170 (4) 775 - 825, 15 March 2021. https://doi.org/10.1215/00127094-2020-0058


Received: 22 March 2019; Revised: 26 June 2020; Published: 15 March 2021
First available in Project Euclid: 15 December 2020

Digital Object Identifier: 10.1215/00127094-2020-0058

Primary: 42B10
Secondary: 22E40 , 37C85

Keywords: additive combinatorics , Fourier transform , Hausdorff and Fourier dimension transfer operators , limit sets of Kleininan groups , Patterson–Sullivan measures , random walks on linear groups , Schottky groups

Rights: Copyright © 2021 Duke University Press


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Vol.170 • No. 4 • 15 March 2021
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