Vol. 11, No. 1, 2022

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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