Real Analysis Exchange

Dimension Spectrum for a Nonconventional Ergodic Average

Yuval Peres and Boris Solomyak

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We compute the dimension spectrum of certain nonconventional averages, namely, the Hausdorff dimension of the set of \(0,1\) sequences, for which the frequency of the pattern 11 in positions \(k, 2k\) equals a given number \(\theta\in [0,1]\).

Article information

Real Anal. Exchange, Volume 37, Number 2 (2011), 375-388.

First available in Project Euclid: 15 April 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A80: Fractals [See also 37Fxx] 37C45: Dimension theory of dynamical systems
Secondary: 28A78: Hausdorff and packing measures

multifractal analysis multiple Birkhoff average Hausdorff dimension


Peres, Yuval; Solomyak, Boris. Dimension Spectrum for a Nonconventional Ergodic Average. Real Anal. Exchange 37 (2011), no. 2, 375--388.

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  • J. Barral and M. Mensi, Multifractal analysis of Birkhoff averages on `self-affine' symbolic spaces, Nonlinearity, 21(10) (2008), 2409–2425.
  • L. Barreira, Dimension and recurrence in hyperbolic dynamics. Progress in Mathematics, 272, Birkhäuser Verlag, Basel, (2008).
  • T. Bedford, Crinkly curves, Markov partitions and box dimension in self-similar sets, Ph.D. Thesis, University of Warwick, (1984).
  • A. S. Besicovitch, On the sum of digits of real numbers represented in the dyadic system, Math. Ann., 110 (1934), 321–330.
  • P. Billingsley, Ergodic theory and information, Wiley, New York, (1965).
  • J. Bourgain, Double recurrence and almost sure convergence, J. Reine Angew. Math., 404 (1990), 140–161.
  • H. G. Eggleston, The fractional dimension of a set defined by decimal properties, Q. J. Math. 20 (1949), 31–36.
  • K. Falconer, Fractal geometry. Mathematical foundations and applications, John Wiley & Sons, Chichester, (1990).
  • A. Fan, L. Liao, J. Ma, Level sets of multiple ergodic averages, Preprint arXiv:1105.3032, to appear in Monatsh. Math.
  • H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Anal. Math. 31 (1977), 204–256.
  • B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds, Ann. Math. 161 (2005), 397–488.
  • T. Jordan and M. Rams, Multifractal analysis for Bedford-McMullen carpets, Math. Proc. Cambridge Philos. Soc. 150 (2011), 147–156.
  • R. Kenyon and Y. Peres, Measures of full dimension on affine-invariant sets, Ergodic Theory Dynam. Systems, 16 (1996), 307–323.
  • R. Kenyon, Y. Peres, and B. Solomyak, Hausdorff dimension of the multiplicative golden mean shift, C. R. Math. Acad. Sci. Paris 349 (2011), 625–628.
  • R. Kenyon, Y. Peres, and B. Solomyak, Hausdorff dimension for fractals invariant under the multiplicative integers, Preprint arXiv 1102.5136, to appear in Ergodic Theory Dynam. Systems.
  • Yu. Kifer, A nonconventional strong law of large numbers and fractal dimensions of some multiple recurrence sets, Preprint arXiv:1012.2799.
  • J. King, The singularity spectrum for general Sierpiński carpets, Adv. Math. 116 (1995), 1–8.
  • R. Lyons, Strong laws of large numbers for weakly correlated random variables, Michigan Math. J. 35 (1988), 353–359.
  • C. McMullen, The Hausdorff dimension of general Sierpinski carpets, Nagoya Math. J. 96 (1984), 1–9.
  • L. Olsen, Self-affine multifractal Sierpinski sponges in $\R^d$, Pacific J. Math. 183(1) (1998), 143–199.