Real Analysis Exchange

On Inhomogeneous Bernoulli Convolutions and Random Power Series

Antonios Bisbas and Jörg Neunhäuserer

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We extend the results of Peres and Solomyak on absolute continuity and singularity of homogeneous Bernoulli convolutions to inhomogeneous ones and generalize the result to random power series given by inhomogeneous Markov chains. In addition we prove an Erd\"{o}s-Salem type theorem for inhomogeneous Bernoulli convolutions.

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Real Anal. Exchange, Volume 36, Number 1 (2010), 213-222.

First available in Project Euclid: 14 March 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A46: Absolutely continuous functions 26A30: Singular functions, Cantor functions, functions with other special properties
Secondary: 28A80: Fractals [See also 37Fxx] 28A78: Hausdorff and packing measures 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure

inhomogeneous Bernoulli convolution random power series absolute continuity singularity Pisot numbers


Bisbas, Antonios; Neunhäuserer, Jörg. On Inhomogeneous Bernoulli Convolutions and Random Power Series. Real Anal. Exchange 36 (2010), no. 1, 213--222.

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  • A. Bisbas and C. Karanikas, On the Hausdorff dimension of Rademacher Riesz products, Monatshefte für Math. 110 (1990), 15–21.
  • A. Bisbas and C. Karanikas, Dimension and entropy of non-ergodic Markovian process and its relation to Rademacher Riez products, Monatshefte für Math. 118 (1994), 21–32.
  • P. Erdös, On a family of symmetric Bernoulli convolutions, Amer. Journ. Math 61 (1939), 974–976.
  • C. Graham and O.C. McGehee, Essays in commutative harmonic analysis, Fundamental principles in Mathematical Science, Springer New York - Berlin (1979).
  • A. Fan and J. Zhang, Absolute continuity of the distribution of some Markov geometric series, Science in China. Series A. Mathematic, 50(11) (2007), 1521-1528.
  • S. Lalley, Random series in powers of algebraic integers: Hausdorff dimension of the limit distribution, Journal of the London Mathematical Society, 57 (1998), 629-654.
  • P. Mattila, Geometry of Sets and Measures in Euclidean spaces, Cambridge University Press (1995).
  • D. Mauldin and K. Simon, The equivalence of some Bernoulli convolution to Lebesgue measure, Proc. of the Amer. Math. Soc. 126(9) (1998), 2733-2736.
  • Y. Peres, W. Schlag and B. Solomyak Sixty years of Bernoulli convolutions, Progress in probability 46 (2000), 39-65.
  • Y. Peres and B. Solomyak, Absolutely continuous Bernoulli convolutions - a simple proof, Math. Research Letters 3(2) (1996), 231-239.
  • Y. Peres and B. Solomyak, Self-similar measures and intersection of Cantor sets, Trans. Amer. Math. Soc 350(10) (1998), 4065-4087.
  • Ya. Pesin, Dimension Theory in Dynamical Systems - Contemplary Views and Applications, University of Chicago Press (1997).
  • R. Salem, Algebraic numbers and Fourier Analysis, Heath (1963).
  • P. Shmerkin and B. Solomyak, Zeros of $\{-1,0,1\}$ power series and connectedness loci of self-affine sets, Experimental Math. 15(4) (2006), 499-511.
  • B. Solomyak, On the random series $\sum \pm \lambda^{i}$ (an Erdös problem), Ann. Math. 142 (1995).