Vol. 9, No. 2, 2020

Download this article
Download this article For screen
For printing
Recent Issues
Volume 13, Issue 2
Volume 13, Issue 1
Volume 12, Issue 4
Volume 12, Issue 3
Volume 12, Issue 2
Volume 12, Issue 1
Volume 11, Issue 4
Volume 11, Issue 3
Volume 11, Issue 2
Volume 11, Issue 1
Volume 10, Issue 4
Volume 10, Issue 3
Volume 10, Issue 2
Volume 10, Issue 1
Volume 9, Issue 4
Volume 9, Issue 3
Volume 9, Issue 2
Volume 9, Issue 1
Volume 8, Issue 4
Volume 8, Issue 3
Volume 8, Issue 2
Volume 8, Issue 1
Older Issues
Volume 7, Issue 4
Volume 7, Issue 3
Volume 7, Issue 2
Volume 7, Issue 1
Volume 6, Issue 4
Volume 6, Issue 2-3
Volume 6, Issue 1
Volume 5, Issue 4
Volume 5, Issue 3
Volume 5, Issue 1-2
Volume 4, Issue 4
Volume 4, Issue 3
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 3-4
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 4
Volume 2, Issue 3
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 4
Volume 1, Issue 3
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
founded and published with the
scientific support and advice of
mathematicians from the
Moscow Institute of
Physics and Technology
Subscriptions
 
ISSN (electronic): 2996-220X
ISSN (print): 2996-2196
Author Index
To Appear
 
Other MSP Journals
This article is available for purchase or by subscription. See below.
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}.

PDF Access Denied

We have not been able to recognize your IP address 3.145.46.4 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

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