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, Tome 35 (2023) no. 1, pp. 1-15.

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.

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.

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DOI : 10.5802/jtnb.1236
Classification : 11A55, 11J20, 11J70, 37B10, 11J71
Mots-clés : 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
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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, Tome 35 (2023) no. 1, pp. 1-15. doi : 10.5802/jtnb.1236. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1236/

[1] Pascal Alessandri; Valérie Berthé Three distance theorems and combinatorics on words, Enseign. Math., Volume 44 (1998) no. 1-2, pp. 103-132 | MR | Zbl

[2] Jean Berstel; Patrice Séébold Sturmian words, Algebraic Combinatorics on Words (Encyclopedia of Mathematics and Its Applications), Volume 90, Cambridge University Press, 2002, pp. 45-110

[3] Kazimierz Florek Une remarque sur la répartition des nombres nξ(mod1), Colloq. Math., Volume 2 (1951), pp. 323-324

[4] John H. Halton The distribution of the sequence {nξ} (n=0,1,2,...), Proc. Camb. Philos. Soc., Volume 61 (1965), pp. 665-670 | DOI | MR

[5] Godfrey H. Hardy; Edward M. Wright An Introduction to the Theory of Numbers, Oxford University Press, 1979

[6] Jeffrey C. Lagarias; Jeffrey O. Shallit Linear fractional transformations of continued fractions with bounded partial quotients, J. Théor. Nombres Bordeaux, Volume 9 (1997) no. 2, pp. 267-279 corrigendum in ibid. 15 (2003), no. 3, p. 741-743 | DOI | Numdam | MR | Zbl

[7] Tony van Ravenstein The three gap theorem (Steinhaus conjecture), J. Aust. Math. Soc., Volume 45 (1988) no. 3, pp. 360-370 | DOI | MR | Zbl

[8] Tony van Ravenstein; Graham Winley; Keith Tognetti Characteristics and the three gap theorem, Fibonacci Q., Volume 28 (1990) no. 3, pp. 204-214 | MR | Zbl

[9] Jeffrey O. Shallit Real numbers with bounded partial quotients, Enseign. Math., Volume 38 (1992) no. 1-2, pp. 151-187 | MR | Zbl

[10] Jeffrey O. Shallit Automaticity IV: Sequences, sets, and diversity, J. Théor. Nombres Bordeaux, Volume 8 (1996) no. 2, pp. 347-367 | DOI | Numdam | MR | Zbl

[11] Noel B. Slater The distribution of the integers N for which {ΘN}<Φ, Proc. Camb. Philos. Soc., Volume 46 (1950), pp. 525-534 | DOI | MR

[12] Vera T. Sós On the theory of diophantine approximations. I, Acta Math. Acad. Sci. Hung., Volume 8 (1957), pp. 461-471 | MR | Zbl

[13] Janos Suranyi Über die Anordnung der Vielfachen einer reellen Zahl mod 1, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math., Volume 1 (1958), pp. 107-111 | Zbl

[14] Stanisław Świerczkowski On successive settings of an arc on the circumference of a circle, Fundam. Math., Volume 46 (1958), pp. 187-189 | DOI | MR

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