Abstract: 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

$\sum_{n\ge0}\pm\left(\lambda_{1}^{n},...,\lambda_{d}^{n}\right)$,

where the $\pm$ are independent fair coin-tosses. Assuming $P(\lambda_{j})
e0$

for all $1\le j\le d$ and nonzero polynomials with coefficients $\pm1,0$,

we show that $\dim\mu_{\lambda}=\min\left\{ d,\dim_{L}\mu_{\lambda}\right\} $,

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.