Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative frequency $p_k$ for all $k \in [0,b-1]$. It is a well known result of H. G. Eggleston that the Hausdorff dimension of $S_{b,v}$ is equal to $$ \frac {-\sum p_k \log p_k} {\log b}. $$

Let $b_1, b_2 \geq 2$ be integers and $v_j=(p_{j,0},p_{j,2},\cdots,p_{j,b_j-1})$ for $j=1,2$ be probability vectors. Clearly, it may happen that $S_{b_1,v_1} \cap S_{b_2,v_2}=\emptyset$ if $b_1=b_2$ and $v_1 \neq v_2$ and under some other conditions, too. Other than trivial statements, is there anything that is known about the Hausdorff dimension of $S_{b_1,v_1} \cap S_{b_2,v_2}$? If not, is it known when this intersection is uncountable or nonempty?