Entropy of Bernoulli convolutions and uniform exponential growth for linear groups

TL;DR

This paper links the entropy of Bernoulli convolutions and the exponential growth of linear groups to the Lehmer conjecture, providing bounds related to algebraic numbers and Mahler measures.

Contribution

It establishes the equivalence between a uniform exponential growth conjecture for linear groups and the Lehmer conjecture, with new bounds on entropy and Bernoulli convolution dimensions.

Findings

Lower bounds for entropy of random walks on algebraic maps in terms of Mahler measure.

Equivalence of exponential growth conjecture and Lehmer conjecture.

Bounds on the dimension of Bernoulli convolutions.

Abstract

The exponential growth rate of non polynomially growing subgroups of $GL_d$ is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture from number theory. In this note, we show that it is equivalent to the Lehmer conjecture. This is done by establishing a lower bound for the entropy of the random walk on the semigroup generated by the maps $x\mapsto \lambda\cdot x\pm 1$, where $\lambda$ is an algebraic number. We give a bound in terms of the Mahler measure of $\lambda$. We also derive a bound on the dimension of Bernoulli convolutions.