The Multiplicative Golden Mean Shift Has Infinite Hausdorff Measure

  • Yuval Peres
  • Boris Solomyak
Part of the Trends in Mathematics book series (TM)


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.


Probability Measure Hausdorff Dimension Binary Sequence Hausdorff Measure Symbolic Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


  1. 1.
    Bedford, T.: Crinkly curves, Markov partitions and box dimension in self-similar sets. Ph.D. Thesis, University of Warwick, (1984)Google Scholar
  2. 2.
    Falconer, K.: Fractal Geometry. Mathematical Foundations and Applications. Wiley, Chichester (1990)MATHGoogle Scholar
  3. 3.
    Fan, A., Liao, L., Ma, J.: Level sets of multiple ergodic averages. Preprint (2011) arXiv:1105.3032. Monatsh. Math. 168(1), 17–26 (2012)Google Scholar
  4. 4.
    Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Statist. Assoc. 58(302), 13–30 (1963)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Kenyon, R., Peres, Y., Solomyak, B.: Hausdorff dimension of the multiplicative golden mean shift. C. R. Math. Acad. Sci. Paris 349, 625–628 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Kenyon, R., Peres, Y., Solomyak, B.: Hausdorff dimension for fractals invariant under the multiplicative integers. Preprint (2011) arXiv 1102.5136. Ergodic Th. Dynam. Sys. 32(5), 1567–1584 (2012)Google Scholar
  7. 7.
    Lalley, S., Gatzouras, D.: Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J. 41(2), 533–568 (1992)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    McMullen, C.: The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J. 96, 1–9 (1984)MathSciNetMATHGoogle Scholar
  9. 9.
    Peres, Y.: The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure. Math. Proc. Camb. Philos. Soc. 116, 513–526 (1994)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Rogers, C.A.: Hausdorff Measures. Cambridge University Press, Cambridge (1970).MATHGoogle Scholar
  11. 11.
    Rogers, C.A., Taylor, S.J.: Functions continuous and singular with respect to a Hausdorff measure. Mathematika 8, 1–31 (1961)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Microsoft ResearchRedmondUSA
  2. 2.Department of MathematicsUniversity of WashingtonUW, SeattleUSA

Personalised recommendations