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Best Diophantine approximations and a multidimensional three distance theorem

Anton Shutov
Vol. 13 (2024), No. 4, 351–359
DOI: 10.2140/cnt.2024.13.351
Abstract

In 1996, N. Chevallier proved a beautiful lemma which connects Diophantine approximation and multidimensional generalizations of the famous three distance theorem. Using this lemma we show how known results about the multidimensional three distance theorem can be deduced from certain known results dealing with the best Diophantine approximations. Also we obtain some new results about the liminf version of the problem. Beside this, we discuss the inverse problem: how results about the multidimensional three distance theorem can be applied to study best Diophantine approximations.

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Keywords
best Diophantine approximations, three distance theorem, Kronecker sequence, Chevallier's lemma
Mathematical Subject Classification
Primary: 11K31
Secondary: 11J13
Milestones
Received: 14 October 2024
Revised: 16 November 2024
Accepted: 30 November 2024
Published: 10 December 2024
Authors
Anton Shutov
Vladimir State University
Vladimir
Russia
© 2024 MSP (Mathematical Sciences Publishers).