Comptes Rendus
no. 7
Number theory/Dynamical systems
Dynamical covering problems on the triadic Cantor set

Presented by: the Editorial Board

Bao-Wei Wang1; Jun Wu1; Jian Xu1
1 School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China
Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 738-743.

In this note, we consider the metric theory of the dynamical covering problems on the triadic Cantor set K. More precisely, let Tx=3x(mod1) be the natural map on K, μ the standard Cantor measure and x0∈K a given point. We consider the size of the set of points in K which can be well approximated by the orbit {Tnx0}n≥1 of x0, namely the set

D(x0,φ):={y∈K:|Tnx0−y|<φ(n)for infinitely manyn∈N},
where φ is a positive function defined on N. It is shown that for μ almost all x0∈K, the Hausdorff measure of D(x0,φ) is either zero or full depending upon the convergence or divergence of a certain series. Among the proof, as a byproduct, we obtain an inhomogeneous counterpart of Levesley, Salp and Velani's work on a Mahler's question about the Diophantine approximation on the Cantor set K.

Nous considérons dans cette Note la théorie métrique des recouvrements dynamiques dans l'ensemble de Cantor triadique K. Plus précisément, soit Tx=3x(mod1) l'application naturelle sur K, μ la mesure de Cantor standard et x0∈K un point donné. Nous considérons la mesure de l'ensemble des points de K qui peuvent être bien approchés par l'orbite {Tnx0}n≥1 de x0, c'est-à-dire l'ensemble

D(x0,φ):={y∈K:|Tnx0−y|<φ(n)pour une infinité den∈N},
où φ est une fonction positive définie sur N. Nous montrons que pour μ-presque tout x0∈K la mesure de Hausdorff de D(x0,φ) est soit zéro, soit pleine, selon la convergence ou la divergence d'une certaine série. Notre démonstration fournit en passant une contre-partie inhomogène au travail de Levesley, Salp et Velani sur une question de Mahler relative à l'approximation rationnelle des points de l'ensemble de Cantor.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.05.014

Bao-Wei Wang 1; Jun Wu 1; Jian Xu 1

1 School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China 
@article{CRMATH_2017__355_7_738_0,
     author = {Bao-Wei Wang and Jun Wu and Jian Xu},
     title = {Dynamical covering problems on the triadic {Cantor} set},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {738--743},
     publisher = {Elsevier},
     volume = {355},
     number = {7},
     year = {2017},
     doi = {10.1016/j.crma.2017.05.014},
     language = {en},
}
TY  - JOUR
AU  - Bao-Wei Wang
AU  - Jun Wu
AU  - Jian Xu
TI  - Dynamical covering problems on the triadic Cantor set
JO  - Comptes Rendus. Mathématique
PY  - 2017
SP  - 738
EP  - 743
VL  - 355
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2017.05.014
LA  - en
ID  - CRMATH_2017__355_7_738_0
ER  - 
%0 Journal Article
%A Bao-Wei Wang
%A Jun Wu
%A Jian Xu
%T Dynamical covering problems on the triadic Cantor set
%J Comptes Rendus. Mathématique
%D 2017
%P 738-743
%V 355
%N 7
%I Elsevier
%R 10.1016/j.crma.2017.05.014
%G en
%F CRMATH_2017__355_7_738_0
Bao-Wei Wang; Jun Wu; Jian Xu. Dynamical covering problems on the triadic Cantor set. Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 738-743. doi : 10.1016/j.crma.2017.05.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.05.014/

[1] J. Barral; S. Seuret Heterogeneous ubiquitous systems in Rd and Hausdorff dimension, Bull. Braz. Math. Soc., Volume 38 (2007) no. 3, pp. 467-515

[2] V. Beresnevich; D. Dickinson; S. Velani Measure Theoretic Laws for Lim Sup Sets, Memoirs of the AMS, vol. 179, 2006 (No. 846, x+91 pp)

[3] V. Beresnevich; S. Velani A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures, Ann. Math. (2), Volume 164 (2006) no. 3, pp. 971-992

[4] Y. Bugeaud Diophantine approximation and Cantor sets, Math. Ann., Volume 341 (2008), pp. 677-684

[5] Y. Bugeaud; A. Durand Metric Diophantine approximation on the middle-third Cantor set, J. Eur. Math. Soc., Volume 18 (2016), pp. 1233-1272

[6] N. Chernov; D. Kleinbock Dynamical Borel–Cantelli lemmas for Gibbs measures, Isr. J. Math., Volume 122 (2001), pp. 1-27

[7] A. Dvoretzky On covering a circle by randomly placed arcs, Proc. Natl. Acad. Sci. USA, Volume 42 (1956), pp. 199-203

[8] A. Fan; J. Schemling; S. Troubetzkoy A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation, Proc. Lond. Math. Soc., Volume 107 (2013) no. 5, pp. 1173-1219

[9] D. Feng; E. Järvenpää; M. Järvenpää; V. Suomala Dimensions of random covering sets in Riemann manifolds, 2015 | arXiv

[10] E. Järvenpää; M. Järvenpää; B. Li; O. Stenflo Random affine code tree fractals and Falcorner–Sloan condition, Ergod. Theory Dyn. Syst., Volume 36 (2016) no. 5, pp. 1516-1533

[11] J.P. Kahane Some Random Series of Functions, Cambridge University Press, Cambridge, UK, 1985

[12] J. Levesley; C. Salp; S. Velani On a problem of K. Mahler: Diophantine approximation and Cantor sets, Math. Ann., Volume 338 (2007), pp. 97-118

[13] L. Liao; S. Seuret Diophantine approximation of orbits in expanding Markov systems, Ergod. Theory Dyn. Syst., Volume 33 (2013) no. 2, pp. 585-608

[14] K. Mahler Some suggestions for further research, Bull. Aust. Math. Soc., Volume 29 (1984), pp. 101-108

[15] W. Philipp Some metrical theorems in number theory, Pac. J. Math., Volume 20 (1967), pp. 109-127

[16] L. Shepp Covering the circle with random arcs, Isr. J. Math., Volume 11 (1972), pp. 328-345

[17] V.G. Sprindz̆uk Metric Theory of Diophantine Approximation, V.H. Winston & Sons, Washington, DC, 1979 (translated by R.A. Silverman)

Cited by Sources:

Comments - Policy