Dimension of a Family of Singular Bernoulli Convolutions

Smoothness of Projections, Bernoulli Convolutions, and the Dimension of Exceptions

YUVAL PERES and WILHELM SCHLAG

On the Smoothness Properties of a Family of Bernoulli Convolutions

Let L (u, a), oo < u < + oo denote the Fourier-Stieltjes transform, 00 f eluoda(x), of a distribution function u(x), oo < x < + co . Thus if 00 /3(x) is the distribution function which is 0, J, 1 according as x -1 < x ,< 1, 1 < x, then L (u,,8) = cos u ; and so, if b is a positive constant, cos (u/b) is the transform of the distribution function /3(bx) . Hence, if a is a positive constant, the ...

Spectra of Bernoulli Convolutions

Sixty Years of Bernoulli Convolutions

The distribution νλ of the random series ∑ ±λ is the infinite convolution product of 1 2 (δ−λn + δλn). These measures have been studied since the 1930’s, revealing connections with harmonic analysis, the theory of algebraic numbers, dynamical systems, and Hausdorff dimension estimation. In this survey we describe some of these connections, and the progress that has been made so far on the funda...

Multifractal Structure of Bernoulli Convolutions

Let ν λ be the distribution of the random series ∑∞ n=1 inλ , where in is a sequence of i.i.d. random variables taking the values 0,1 with probabilities p, 1 − p. These measures are the well-known (biased) Bernoulli convolutions. In this paper we study the multifractal spectrum of ν λ for typical λ. Namely, we investigate the size of the sets ∆λ,p(α) = { x ∈ R : lim r↘0 log ν λ(B(x, r)) log r =...