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The Multiplicative Golden Mean Shift Has Infinite Hausdorff Measure

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Further Developments in Fractals and Related Fields

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Abstract

In an earlier work, joint with R. Kenyon, we computed 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 2k  = 0 for all k ≥ 1. Here we show that this set has infinite Hausdorff measure in its dimension. A more precise result in terms of gauges in which the Hausdorff measure is infinite is also obtained.

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Acknowledgements

The research of B. S. was supported in part by the NSF grant DMS-0968879. He is grateful to the Microsoft Research Theory Group for hospitality during 2010–2011. He would also like to thank the organizers of the conference “Fractals and Related Fields II” for the excellent meeting and stimulating atmosphere. The authors are grateful to the referee for careful reading of the manuscript and many helpful comments.

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Authors and Affiliations

  1. Microsoft Research, One Microsoft Way, Redmond, WA, 98052, USA

    Yuval Peres

  2. Department of Mathematics, University of Washington, 354350, UW, Seattle, WA, 98195, USA

    Boris Solomyak

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  1. Yuval Peres
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  2. Boris Solomyak
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Correspondence to Boris Solomyak .

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Editors and Affiliations

  1. , LAGA, Université Paris 13, avenue J.B. Clément 99, Villetaneuse, 93430, France

    Julien Barral

  2. , Laboratoire d'Analyse et de Mathématique, Université Paris-Est Créteil - Val de Ma, Avenue du Général de Gaulle 61, Créteil Cedex, 94010, France

    Stéphane Seuret

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Peres, Y., Solomyak, B. (2013). The Multiplicative Golden Mean Shift Has Infinite Hausdorff Measure. In: Barral, J., Seuret, S. (eds) Further Developments in Fractals and Related Fields. Trends in Mathematics. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8400-6_10

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