The Michigan Mathematical Journal

Cubic relations between frequencies of digits and Hausdorff dimension

Luis Barreira and Claudia Valls

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Michigan Math. J., Volume 60, Issue 3 (2011), 579-602.

First available in Project Euclid: 8 November 2011

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Zentralblatt MATH identifier

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension [See also 11N99, 28Dxx] 37C45: Dimension theory of dynamical systems


Barreira, Luis; Valls, Claudia. Cubic relations between frequencies of digits and Hausdorff dimension. Michigan Math. J. 60 (2011), no. 3, 579--602. doi:10.1307/mmj/1320763050.

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  • J. Barral and S. Seuret, Ubiquity and large intersections properties under digit frequencies constraints, Math. Proc. Cambridge Philos. Soc. 145 (2008), 527–548.
  • L. Barreira, Dimension and recurrence in hyperbolic dynamics, Progr. Math., 272, Birkhäuser, Basel, 2008.
  • L. Barreira, B. Saussol, and J. Schmeling, Distribution of frequencies of digits via multifractal analysis, J. Number Theory 97 (2002), 413–442.
  • –––, Higher-dimensional multifractal analysis, J. Math. Pures Appl. (9) 81 (2002), 67–91.
  • L. Barreira and C. Valls, Asymptotic behavior of distribution of frequencies of digits, Math. Proc. Cambridge Philos. Soc. 145 (2008), 177–195.
  • A. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1935), 321–330.
  • P. Billingsley, Ergodic theory and information, Wiley, New York, 1965.
  • G. Constantine and T. Savits, A multivariate Faà di Bruno formula with applications, Trans. Amer. Math. Soc. 348 (1996), 503–520.
  • A. Durand, Ubiquitous systems and metric number theory, Adv. Math. 218 (2008), 368–394.
  • –––, Large intersection properties in Diophantine approximation and dynamical systems, J. London Math. Soc. (2) 79 (2009), 377–398.
  • H. Eggleston, The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser. 20 (1949), 31–36.
  • A. Fan, D. Feng, and J. Wu, Recurrence, dimension and entropy, J. London Math. Soc. (2) 64 (2001), 229–244.
  • Ya. Pesin, Dimension theory in dynamical systems: Contemporary views and applications, Chicago Lectures in Math., Univ. Chicago Press, 1997.
  • F. Takens and E. Verbitskiy, On the variational principle for the topological entropy of certain non-compact sets, Ergodic Theory Dynam. Systems 23 (2003), 317–348.