Vol. 11, No. 1, 2022

Download this article
Download this article For screen
For printing
Recent Issues
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
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
founded and published with the
scientific support and advice of
mathematicians from the
Moscow Institute of
Physics and Technology
 
ISSN (electronic): 2640-7361
ISSN (print): 2220-5438
Author Index
To Appear
 
Other MSP Journals
This article is available for purchase or by subscription. See below.
Abundance of Dirichlet-improvable pairs with respect to arbitrary norms

Dmitry Kleinbock and Anurag Rao

Vol. 11 (2022), No. 1, 97–114
Abstract

Akhunzhanov and Shatskov (Mosc. J. Comb. Number Theory 3:3-4 (2013), 5–23) defined the two-dimensional Dirichlet spectrum with respect to Euclidean norm. We consider an analogous definition for arbitrary norms on 2 and prove that, for each such norm, the set of Dirichlet-improvable pairs contains the set of badly approximable pairs, and hence is hyperplane absolute winning. To prove this we make a careful study of some classical results in the geometry of numbers due to Chalk–Rogers and Mahler to establish a Hajós–Minkowski-type result for the critical locus of a cylinder. As a corollary, using a recent result of Kleinbock and Mirzadeh (arXiv:2010.14065 (2020)), we conclude that for any norm on 2 the top of the Dirichlet spectrum is not an isolated point.

PDF Access Denied

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/moscow

We have not been able to recognize your IP address 44.201.96.43 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
Dirichlet's theorem, geometry of numbers, critical lattices
Mathematical Subject Classification
Primary: 11J13
Secondary: 11J83, 11H06, 37A17
Milestones
Received: 26 October 2021
Revised: 29 January 2022
Accepted: 13 February 2022
Published: 30 March 2022
Authors
Dmitry Kleinbock
Department of Mathematics
Brandeis University
Waltham, MA
United States
Anurag Rao
Department of Mathematics and Computer Science
Wesleyan University
Middletown, CT
United States