## Real Analysis Exchange

### On Inhomogeneous Bernoulli Convolutions and Random Power Series

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 14 March 2011

https://projecteuclid.org/euclid.rae/1300108094

Mathematical Reviews number (MathSciNet)
MR3016413

Zentralblatt MATH identifier
1247.26017

#### Citation

Bisbas, Antonios; Neunhäuserer, Jörg. On Inhomogeneous Bernoulli Convolutions and Random Power Series. Real Anal. Exchange 36 (2010), no. 1, 213--222. https://projecteuclid.org/euclid.rae/1300108094

#### References

• 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).