On the dimension of Bernoulli convolutions

TL;DR

This paper investigates the dimensions of Bernoulli convolutions, showing how parameters with less than full dimension can be approximated by algebraic numbers, and providing explicit examples of transcendental parameters with full dimension.

Contribution

It establishes approximation results for parameters with less than full dimension and identifies explicit transcendental parameters with full dimension, linking to Lehmer's conjecture.

Findings

Parameters with less than full dimension can be approximated by algebraic parameters with controlled error.

Explicit transcendental parameters with full dimension are identified.

Lehmer's conjecture implies a positive measure interval of parameters with full dimension.

Abstract

The Bernoulli convolution with parameter $\lambda\in(0,1)$ is the probability measure $\mu_\lambda$ that is the law of the random variable $\sum_{n\ge0}\pm\lambda^n$, where the signs are independent unbiased coin tosses. We prove that each parameter $\lambda\in(1/2,1)$ with $\dim\mu_\lambda<1$ can be approximated by algebraic parameters $\xi\in(1/2,1)$ within an error of order $\exp(-deg(\xi)^{A})$ for any number $A$, such that $\dim\mu_\xi<1$. As a corollary, we conclude that $\dim\mu_\lambda=1$ for each of $\lambda=\ln 2, e^{-1/2}, \pi/4$. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conjecture implies the existence of a constant $a<1$ such that $\dim\mu_\lambda=1$ for all $\lambda\in(a,1)$.