The dimension of Bernoulli convolutions in $\mathbb{R}^d$

Abstract: For $(\lambda_{1},\dots,\lambda_{d})=\lambda\in(0,1)^{d}$ with $\lambda_{1}>\cdots>\lambda_{d}$, denote by $\mu_{\lambda}$ the Bernoulli convolution associated to $\lambda$. That is, $\mu_{\lambda}$ is the distribution of the random vector $\sum_{n\ge0}\pm\left(\lambda_{1}^{n},\dots,\lambda_{d}^{n}\right)$, 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\{ \dim_{L}\mu_{\lambda},d\} $, where $\dim_{L}\mu_{\lambda}$ is the Lyapunov dimension. This is a joint work with Ariel Rapaport.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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