Uniform lower bounds on the dimension of Bernoulli convolutions

TL;DR

This paper introduces an algorithm to establish a uniform lower bound on the Hausdorff dimension of Bernoulli convolutions, demonstrating that for all parameters between 0.5 and 1, the dimension is at least approximately 0.964.

Contribution

The authors develop a new algorithm that provides a uniform lower bound on the Hausdorff dimension of Bernoulli convolutions across a range of parameters.

Findings

Uniform lower bound of 0.96399 for all 0.5<λ<1

Algorithm applicable to affine iterated function schemes with similarities

Improves understanding of the dimension spectrum 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 $\frac12 < \lambda < 1$. In particular, for Bernoulli convolutions we give a uniform lower bound $\dim_H(\mu_\lambda) \geq 0.96399$ for all $\frac12<\lambda<1$.