Hausdorff dimension of the multiplicative golden mean shift

Richard Kenyon; Yuval Peres; Boris Solomyak

doi:10.1016/j.crma.2011.05.009

Mathematical Analysis/Dynamical Systems

Hausdorff dimension of the multiplicative golden mean shift

Presented by: Jean-Pierre Kahane

Richard Kenyon ¹; Yuval Peres ²; Boris Solomyak ³

¹Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence, RI 02912, USA
²Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA
³Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA

Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 625-628

Abstract (Original language)
Résumé

We compute the Hausdorff dimension of the “multiplicative golden mean shift” defined as the set of all reals in $[0, 1]$ whose binary expansion $(x_{k})$ satisfies $x_{k} x_{2 k} = 0$ for all $k ⩾ 1$ , 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 $(x_{k})$ satisfait $x_{k} x_{2 k} = 0$ pour tout $k ⩾ 1$ . Nous montrons que la dimension de Hausdorff est plus petite que la dimension de Minkowski.

Received: 2011-05-02
Accepted: 2011-05-17
Published online: 2011-06-01

DOI: 10.1016/j.crma.2011.05.009

Author's affiliations:

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

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     title = {Hausdorff dimension of the multiplicative golden mean shift},
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     language = {en},
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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

References
Cited by

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