Uniform lower bounds on the dimension of Bernoulli convolutions

Abstract

In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The Bernoulli convolution measure ${\mu }_{\lambda }$ is the probability measure corresponding to the law of the random variable $\xi ={\sum }_{k=0}^{\infty }{\xi }_k{\lambda }^k$, where ${\xi }_k$ are i.i.d. random variables assuming values $-1$ and $1$ with equal probability and $\frac{1}{2}<\lambda <1$. In particular, for Bernoulli convolutions we give a uniform lower bound ${\dim }_H({\mu }_{\lambda })\ge 0.96399$ for all $\frac{1}{2}<\lambda <1$.

arXiv comment50 pages, 11 figures