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Dimensions of some fractals defined via the semigroup generated by 2 and 3

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Abstract

We compute the Hausdorff and Minkowski dimension of subsets of the symbolic space Σ m ={0, ...,m−1}ℕ that are invariant under multiplication by integers. The results apply to the sets {x∈Σ m :∀ k, x k x 2k ... x nk =0}, where n ≥ 3. We prove that for such sets, the Hausdorff and Minkowski dimensions typically differ.

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Authors and Affiliations

  1. Microsoft Research, One Microsoft Way, Redmond, WA, 981052, USA

    Yuval Peres

  2. Centre for Mathematial Sciences, Lund Institute of Technology/Lund University, Box 118, SE-22100, Line, Sweden

    Joerg Schmeling

  3. Department of Mathematics, LAMA UMR CNRS 8050, Université Paris-Est Créteil, 940010, Créteil Cedex, France

    Stéphane Seuret

  4. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195, USA

    Boris Solomyak

Authors
  1. Yuval Peres
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  2. Joerg Schmeling
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  3. Stéphane Seuret
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  4. Boris Solomyak
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Correspondence to Yuval Peres.

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Peres, Y., Schmeling, J., Seuret, S. et al. Dimensions of some fractals defined via the semigroup generated by 2 and 3. Isr. J. Math. 199, 687–709 (2014). https://doi.org/10.1007/s11856-013-0058-z

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  • DOI: https://doi.org/10.1007/s11856-013-0058-z

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