Dimension of Bernoulli Convolutions in $\mathbb{R}^{d}$

TL;DR

This paper establishes the dimension of Bernoulli convolutions in higher dimensions under certain algebraic conditions, extending previous one-dimensional results and introducing a new entropy increase estimate for non-conformal systems.

Contribution

It extends the dimension results of Bernoulli convolutions to higher dimensions and non-conformal systems, with a novel explicit entropy increase estimate.

Findings

Dimension of $mbda$ equals the minimum of Lyapunov dimension and $d$.

Results hold for $mbda$ with non-root polynomial conditions.

Extension of entropy increase results to higher-dimensional non-conformal systems.

Abstract

For $(\lambda_{1},...,\lambda_{d})=\lambda\in(0,1)^{d}$ with $\lambda_{1}>...>\lambda_{d}$, denote by $\mu_{\lambda}$ the Bernoulli convolution associated to $\lambda$. That is, $\mu_{\lambda}$ is the distribution of the random vector $\sum_{n\ge0}\pm\left(\lambda_{1}^{n},...,\lambda_{d}^{n}\right)$, where the $\pm$ signs are chosen independently and with equal weight. Assuming for each $1\le j\le d$ that $\lambda_{j}$ is not a root of a polynomial with coefficients $\pm1,0$, we prove that the dimension of $\mu_{\lambda}$ equals $\min\left\{ \dim_{L}\mu_{\lambda},d\right\} $, where $\dim_{L}\mu_{\lambda}$ is the Lyapunov dimension. More generally, we obtain this result in the context of homogeneous diagonal self-affine systems on $\mathbb{R}^{d}$ with rational translations. The proof extends to higher dimensions the works of Breuillard and Varj\'u and Varj\'u regarding Bernoulli convolutions on the real line. The main novelty and contribution of the present work lies in an extension of an entropy increase result, due to Varj\'u, in which the amount of increase in entropy is given explicitly. The extension of this result to the higher-dimensional non-conformal case requires significant new ideas.