Badly approximable numbers, Kronecker’s theorem, and diversity of Sturmian characteristic sequences

Badziahin, Dmitry; Shallit, Jeffrey

Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 1, pp. 1-15

We give an optimal version of the classical “three-gap theorem” on the fractional parts of $n\theta$, in the case where $\theta$ 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 $n\theta$, dans le cas où $\theta$ 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.

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

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