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

Abstract

For $(λ_{1},...,λ_{d})=λ\in(0,1)^{d}$ with $λ_{1}>...>λ_{d}$, denote by $μ_λ$ the Bernoulli convolution associated to $λ$. That is, $μ_λ$ is the distribution of the random vector $\sum_{n\ge0}\pm\left(λ_{1}^{n},...,λ_{d}^{n}\right)$, where the $\pm$ signs are chosen independently and with equal weight. Assuming for each $1\le j\le d$ that $λ_{j}$ is not a root of a polynomial with coefficients $\pm1,0$, we prove that the dimension of $μ_λ$ equals $\min\left\{ \dim_{L}μ_λ,d\right\} $, where $\dim_{L}μ_λ$ 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 Varjú and Varjú 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 Varjú, 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.