Dimension of Bernoulli convolutions in R^d

Let $(\lambda_{1},…,\lambda_{d})=\lambda\in(0,1)^{d}$ be with $\lambda_{1}>…>\lambda_{d}$ and let $\mu_{\lambda}$ be the distribution of the random vector $\sum_{n\ge0}\pm (\lambda_{1}^{n},…,\lambda_{d}^{n})$, where the $\pm$ are independent fair coin-tosses. Assuming $P(\lambda_{j})\ne 0$ for all $1\le j\le d$ and nonzero polynomials with coefficients $\pm1,0$, we show that $\operatorname{dim}\mu_{\lambda}=\min \big(d,\dim_{L}\mu_{\lambda} \big)$, where $\dim_{L}\mu_{\lambda}$ is the Lyapunov dimension. This extends to higher dimensions a result of Varjú from 2018 regarding the dimension of Bernoulli convolutions on the real line. Joint work with Haojie Ren.