Archiv der Mathematik

, Volume 102, Issue 3, pp 283–291 | Cite as

Geometric series with randomly increasing exponents

  • J. Neunhäuserer


We prove results on absolute continuity and singularity of the distribution of geometric series with randomly increasing exponents.

Mathematics Subject Classification (2010)

Primary 26A46 26A30 Secondary 28A78 28A80 


Random geometric series Singularity Absolute continuity Hausdorff dimension Density 


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  1. 1.
    Björklund M., Schnellmann D.: Almost sure absolute continuity of Bernoulli convolutions. Ann. Inst. H. Poincaré Probab. Statist. 46, 888–893 (2010)CrossRefMATHGoogle Scholar
  2. 2.
    P. Billingsley, Ergodic Theory and Information, John Wiley and Sons, 1965.Google Scholar
  3. 3.
    N. Bourbaki, Elements of Mathematics: General Topology, Addison-Wesley, 1966.Google Scholar
  4. 4.
    Dajani K, Fieldsteel A: Equipartition of interval partitions and an application to number theory. Proc. Amer. Math. Soc. 129, 3453–3460 (2001)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    K. Falconer, Fractal Geometry - mathematical foundations and applications, John Wiley and Sons, 1990.Google Scholar
  6. 6.
    A. Fan and J. Zhang, Absolute continuity of the distribution of some Markov geometric series, Science in China, Series A Mathematics, vol. 50 (2007), 1521–1528.Google Scholar
  7. 7.
    H. Fernau, Infinite Iterated function systems, Mathematische Nachrichten, vol. 170 (1994), 79–91.Google Scholar
  8. 8.
    Hutchinson J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 271–280 (1981)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Jordan T, Pollicott M, Simon K: Hausdorff dimension of randomly perturbed self-affine attractors. Comm. in Math. Phys. 270, 519–544 (2007)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge University Press, 1995.Google Scholar
  11. 11.
    J. Neunhäuserer, Properties of some overlapping self-similar and some self-affine measures, Acta Mathematica Hungarica, vol. 92 (2001), 143–161.Google Scholar
  12. 12.
    Neunhäuserer J.: A general result on absolute continuity of non-uniform self-similar measures on the real line. Fractals 16, 299–304 (2008)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Y. Peres, W. Schlag, and B. Solomyak, Sixty years of Bernoulli convolutions, Progress in Probability Vol. 46, Birkhauser, 39–65.Google Scholar
  14. 14.
    Peres Y, Solomyak B: Absolutely continuous Bernoulli convolutions - a simple proof. Math. Research Letters 3, 231–239 (1996)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Peres Y, Solomyak B: Self-similar measures and intersection of Cantor sets, Trans. Amer. Math. Soc 350, 4065–4087 (1998)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Ya. Pesin, Dimension Theory in Dynamical Systems - Contemporary Views and Applications, University of Chicago Press, 1997.Google Scholar
  17. 17.
    K. Simon and H. Toth, The absolute continuity of the distribution of random sums with digits {0,1, . . . , m−1}, Real Analysis Exchange, vol 30 (2005), 397–410.Google Scholar
  18. 18.
    Shmerkin P, Solomyak B: Zeros of {−1, 0, 1} power series and connectedness loci of self-affine sets. Experiment. Math. 15, 499–511 (2006)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Toth H.: Infinite Bernoulli convolutions with different probabilities. Dis. Contin. Dyn. Sys. 21, 595–600 (2008)CrossRefMATHGoogle Scholar
  20. 20.
    Young L.S.: Dimension, entropy and Lyapunov exponents. Ergod. Thy. & Dynam. Sys. 2, 109–124 (1982)CrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.GoslarGermany

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