Badly approximable numbers, Kronecker’s theorem, and diversity of Sturmian characteristic sequences
Dmitry Badziahin1; Jeffrey Shallit2
1 School of Mathematics and Statistics University of Sydney NSW 2006 Australia
2 School of Computer Science University of Waterloo Waterloo, ON N2L 3G1 Canada
Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 1-15.

We give an optimal version of the classical “three-gap theorem” on the fractional parts of nθ, in the case where θ is an irrational number that is badly approximable. As a consequence, we deduce a version of Kronecker’s inhomogeneous approximation theorem in one dimension for badly approximable numbers. We apply these results to obtain an improved measure of sequence diversity for characteristic Sturmian sequences, where the slope is badly approximable.

Nous donnons une version optimale du théorème classique des “trois distances” concernant les parties fractionnaires de nθ, dans le cas où θ est un nombre irrationnel qui est mal approchable. Comme conséquence, nous obtenons une version du théorème d’approximation inhomogène de Kronecker, en une dimension, pour les nombres mal approchables. Nous appliquons ces résultats à l’obtention d’une mesure améliorée de la “diversité” des suites sturmiennes caractéristiques dont la pente est mal approchable.

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Accepted:
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DOI: 10.5802/jtnb.1236
Classification: 11A55, 11J20, 11J70, 37B10, 11J71
Keywords: badly approximable number, bounded partial quotients, continued fraction, Kronecker’s theorem, Sturmian characteristic sequence, three-gap theorem, measure of diversity
Dmitry Badziahin 1; Jeffrey Shallit 2

1 School of Mathematics and Statistics University of Sydney NSW 2006 Australia
2 School of Computer Science University of Waterloo Waterloo, ON N2L 3G1 Canada
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Dmitry Badziahin; Jeffrey Shallit. Badly approximable numbers, Kronecker’s theorem, and diversity of Sturmian characteristic sequences. Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 1-15. doi : 10.5802/jtnb.1236. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1236/

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