Comptes Rendus
no. 4
Number theory
Exceptional parameters of linear mod one transformations and fractional parts {ξ(p/q)n}
[Paramètres exceptionnels de transformations linéaires mod un et parties fractionnaires {ξ(p/q)n}]

Présenté par : the Editorial Board

DoYong Kwon1
1 Department of Mathematics, Chonnam National University, Gwangju 500-757, Republic of Korea
Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 291-296.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.01.017
DoYong Kwon 1

1 Department of Mathematics, Chonnam National University, Gwangju 500-757, Republic of Korea 
@article{CRMATH_2015__353_4_291_0,
     author = {DoYong Kwon},
     title = {Exceptional parameters of linear mod one transformations and fractional parts $ \{\xi {(p/q)}^{n}\}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {291--296},
     publisher = {Elsevier},
     volume = {353},
     number = {4},
     year = {2015},
     doi = {10.1016/j.crma.2015.01.017},
     language = {en},
}
TY  - JOUR
AU  - DoYong Kwon
TI  - Exceptional parameters of linear mod one transformations and fractional parts $ \{\xi {(p/q)}^{n}\}$
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 291
EP  - 296
VL  - 353
IS  - 4
PB  - Elsevier
DO  - 10.1016/j.crma.2015.01.017
LA  - en
ID  - CRMATH_2015__353_4_291_0
ER  - 
%0 Journal Article
%A DoYong Kwon
%T Exceptional parameters of linear mod one transformations and fractional parts $ \{\xi {(p/q)}^{n}\}$
%J Comptes Rendus. Mathématique
%D 2015
%P 291-296
%V 353
%N 4
%I Elsevier
%R 10.1016/j.crma.2015.01.017
%G en
%F CRMATH_2015__353_4_291_0
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/

[1] Y. Bugeaud Linear mod one transformations and the distribution of fractional parts {ξ(p/q)n}, Acta Arith., Volume 114 (2004) no. 4, pp. 301-311

[2] Y. Bugeaud; A. Dubickas Fractional parts of powers and Sturmian words, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 341 (2005) no. 2, pp. 69-74

[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

[5] L. Flatto; J.C. Lagarias; A.D. Pollington On the range of fractional parts {ξ(p/q)n}, Acta Arith., Volume 70 (1995) no. 2, pp. 125-147

[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

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Quasicrystals, model sets, and automatic sequences

Jean-Paul Allouche; Yves Meyer

C. R. Phys (2014)