Comptes Rendus
Number theory
Exceptional parameters of linear mod one transformations and fractional parts {ξ(p/q)n}
Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 291-296.

We study exceptional parameters of linear mod one transformations. The present note proves that the set of such values has Hausdorff dimension zero. This answers the question posed by Bugeaud.

Nous étudions des paramètres exceptionnels de transformations linéaires mod un. La présente note prouve que l'ensemble de ces valeurs a zéro pour dimension de Hausdorff. Ceci répond à la question posée par Bugeaud.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2015.01.017
DoYong Kwon 1

1 Department of Mathematics, Chonnam National University, Gwangju 500-757, Republic of Korea
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DoYong Kwon. Exceptional parameters of linear mod one transformations and fractional parts $ \{\xi {(p/q)}^{n}\}$. Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 291-296. doi : 10.1016/j.crma.2015.01.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.01.017/

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[3] A. Dubickas Powers of a rational number modulo 1 cannot lie in a small interval, Acta Arith., Volume 137 (2009) no. 3, pp. 233-239

[4] K. Falconer Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, Inc., Hoboken, NJ, USA, 2003

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[6] G.H. Hardy; E.M. Wright An Introduction to the Theory of Numbers, Oxford University Press, Oxford, 2008

[7] D.Y. Kwon A devil's staircase from rotations and irrationality measures for Liouville numbers, Math. Proc. Camb. Philos. Soc., Volume 145 (2008) no. 3, pp. 739-756

[8] D.Y. Kwon The natural extensions of β-transformations which generalize Baker's transformations, Nonlinearity, Volume 22 (2009) no. 2, pp. 301-310

[9] D.Y. Kwon The orbit of a β-transformation cannot lie in a small interval, J. Korean Math. Soc., Volume 49 (2012) no. 4, pp. 867-879

[10] D.Y. Kwon Moments of discrete measures with dense jumps induced by β-expansions, J. Math. Anal. Appl., Volume 399 (2013) no. 1, pp. 1-11

[11] D.Y. Kwon A two-dimensional singular function via Sturmian words in base β, J. Number Theory, Volume 133 (2013) no. 11, pp. 3982-3994

[12] M. Lothaire Algebraic Combinatorics on Words, Cambridge University Press, Cambridge, 2002

[13] K. Mahler An unsolved problem on the powers of 3/2, J. Aust. Math. Soc., Volume 8 (1968), pp. 313-321

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