On the dimension of Bernoulli convolutions for all transcendental parameters

TL;DR

This paper proves that Bernoulli convolutions have full dimension for all transcendental parameters in (1/2,1), advancing understanding of their fractal structure.

Contribution

It establishes that the dimension of Bernoulli convolutions equals 1 for all transcendental parameters in (1/2,1), a significant extension of previous results.

Findings

Dimension of Bernoulli convolutions is 1 for all transcendental parameters in (1/2,1)

Supports conjectures about the fractal nature of these measures

Provides new techniques for analyzing transcendental parameters

Abstract

The Bernoulli convolution $\nu_\lambda$ with parameter $\lambda\in(0,1)$ is the probability measure supported on $\mathbf{R}$ that is the law of the random variable $\sum\pm\lambda^n$, where the $\pm$ are independent fair coin-tosses. We prove that $\dim\nu_\lambda=1$ for all transcendental $\lambda\in(1/2,1)$.