Comptes Rendus
Mathematical Analysis/Dynamical Systems
Hausdorff dimension of the multiplicative golden mean shift
Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 625-628.

We compute the Hausdorff dimension of the “multiplicative golden mean shift” defined as the set of all reals in [0,1] whose binary expansion (xk) satisfies xkx2k=0 for all k1, and show that it is smaller than the Minkowski dimension.

Nous calculons la dimension de Hausdorff du « shift de Fibonacci multiplicatif », cʼest-à-dire lʼensemble des nombres réels dans [0,1] dont le développement en binaire (xk) satisfait xkx2k=0 pour tout k1. Nous montrons que la dimension de Hausdorff est plus petite que la dimension de Minkowski.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.05.009

Richard Kenyon 1; Yuval Peres 2; Boris Solomyak 3

1 Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, RI 02912, USA
2 Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA
3 Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA
@article{CRMATH_2011__349_11-12_625_0,
     author = {Richard Kenyon and Yuval Peres and Boris Solomyak},
     title = {Hausdorff dimension of the multiplicative golden mean shift},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {625--628},
     publisher = {Elsevier},
     volume = {349},
     number = {11-12},
     year = {2011},
     doi = {10.1016/j.crma.2011.05.009},
     language = {en},
}
TY  - JOUR
AU  - Richard Kenyon
AU  - Yuval Peres
AU  - Boris Solomyak
TI  - Hausdorff dimension of the multiplicative golden mean shift
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 625
EP  - 628
VL  - 349
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2011.05.009
LA  - en
ID  - CRMATH_2011__349_11-12_625_0
ER  - 
%0 Journal Article
%A Richard Kenyon
%A Yuval Peres
%A Boris Solomyak
%T Hausdorff dimension of the multiplicative golden mean shift
%J Comptes Rendus. Mathématique
%D 2011
%P 625-628
%V 349
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2011.05.009
%G en
%F CRMATH_2011__349_11-12_625_0
Richard Kenyon; Yuval Peres; Boris Solomyak. Hausdorff dimension of the multiplicative golden mean shift. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 625-628. doi : 10.1016/j.crma.2011.05.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.05.009/

[1] T. Bedford, Crinkly curves, Markov partitions and box dimension in self-similar sets, PhD thesis, University of Warwick, 1984.

[2] P. Billingsley Ergodic Theory and Information, Wiley, New York, 1965

[3] K. Falconer Techniques in Fractal Geometry, John Wiley & Sons, Chichester, 1997

[4] A. Fan; L. Liao; J. Ma Level sets of multiple ergodic averages (preprint) | arXiv

[5] H. Furstenberg Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory, Volume 1 (1967), pp. 1-49

[6] R. Kenyon; Y. Peres; B. Solomyak Hausdorff dimension for fractals invariant under the multiplicative integers, 2011 (preprint) | arXiv

[7] C. McMullen The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J., Volume 96 (1984), pp. 1-9

Cited by Sources:

Comments - Policy