Vol. 11, No. 1, 2022

Download this article
Download this article For screen
For printing
Recent Issues
Volume 13, Issue 4
Volume 13, Issue 3
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 2996-220X (online)
ISSN 2996-2196 (print)
Author Index
To Appear
 
Other MSP Journals
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.

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