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Counterexamples to Minkowski's conjecture and escape of mass in positive characteristic

Noy Soffer Aranov

Vol. 13 (2024), No. 3, 179–205
Abstract

We show that there are infinitely many counterexamples to Minkowski’s conjecture in positive characteristic regarding uniqueness of the upper bound of the multiplicative covering radius, μ, by constructing a sequence of compact A-orbits where μ obtains its conjectured upper bound. In addition, we show that these orbits, as well as a slightly larger sequence of orbits, must exhibit complete escape of mass.

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Keywords
compact orbit, diagonal group, function fields, positive characteristic, Minkowski conjecture, geometry of numbers, covering radius, measures, escape of mass
Mathematical Subject Classification
Primary: 11H46
Secondary: 37A44
Milestones
Received: 4 March 2023
Revised: 27 February 2024
Accepted: 30 April 2024
Published: 16 August 2024
Authors
Noy Soffer Aranov
Department of Mathematics
Technion
Haifa
Israel