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Abstract
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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,
, by constructing a
sequence of compact
-orbits
where
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
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Mathematical Subject Classification
Primary: 11H46
Secondary: 37A44
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Milestones
Received: 4 March 2023
Revised: 27 February 2024
Accepted: 30 April 2024
Published: 16 August 2024
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| © 2024 The Author(s), under
exclusive license to MSP (Mathematical Sciences
Publishers). |
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