Vol. 9, No. 2, 2020

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A dynamical Borel–Cantelli lemma via improvements to Dirichlet's theorem

Dmitry Kleinbock and Shucheng Yu

Vol. 9 (2020), No. 2, 101–122
Abstract

Let XSL2()SL2() be the space of unimodular lattices in 2, and for any r 0 denote by Kr X 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 r 0. 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}.

Keywords
Siegel transform, dynamical Borel–Cantelli lemma
Mathematical Subject Classification 2010
Primary: 11J04, 37A17
Secondary: 11H60, 37D40
Milestones
Received: 2 October 2019
Revised: 30 December 2019
Accepted: 14 January 2020
Published: 29 February 2020
Authors
Dmitry Kleinbock
Department of Mathematics
Brandeis University
Waltham, MA
United States
Shucheng Yu
Department of Mathematics
Technion
Haifa
Israel