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2020 A dynamical Borel–Cantelli lemma via improvements to Dirichlet's theorem
Dmitry Kleinbock, Shucheng Yu
Mosc. J. Comb. Number Theory 9(2): 101-122 (2020). DOI: 10.2140/moscow.2020.9.101

Abstract

Let XSL2()SL2() be the space of unimodular lattices in 2, and for any r0 denote by KrX the set of lattices such that all its nonzero vectors have supremum norm at least er. These are compact nested subsets of X, with K0=rKr being the union of two closed horocycles. We use an explicit second moment formula for the Siegel transform of the indicator functions of squares in 2 centered at the origin to derive an asymptotic formula for the volume of sets Kr as r0. Combined with a zero-one law for the set of the ψ-Dirichlet numbers established by Kleinbock and Wadleigh (Proc. Amer. Math. Soc. 146 (2018), 1833–1844), this gives a new dynamical Borel–Cantelli lemma for the geodesic flow on X with respect to the family of shrinking targets {Kr}.

Citation

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Dmitry Kleinbock. Shucheng Yu. "A dynamical Borel–Cantelli lemma via improvements to Dirichlet's theorem." Mosc. J. Comb. Number Theory 9 (2) 101 - 122, 2020. https://doi.org/10.2140/moscow.2020.9.101

Information

Received: 2 October 2019; Revised: 30 December 2019; Accepted: 14 January 2020; Published: 2020
First available in Project Euclid: 17 September 2020

zbMATH: 07211442
MathSciNet: MR4096116
Digital Object Identifier: 10.2140/moscow.2020.9.101

Subjects:
Primary: 11J04, 37A17
Secondary: 11H60, 37D40

Rights: Copyright © 2020 Mathematical Sciences Publishers

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