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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?

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Just a comment: this is quite closely related to the Furstenberg conjecture in ergodic theory, which asks for a classification of measures simultaneously invariant under multiplication by 2 and by 3. – Anthony Quas Jul 26 '13 at 16:19
I heard an interesting talk by Ted Slaman the other week about normal numbers. In particular, apart from trivial conditions, normality in one base is essentially completely independent from normality in another. Do these methods (which involve looking at measures on certain - not necessarily self-similar - Cantor sets) not give an approach to solving this problem? – Lasse Rempe-Gillen Oct 16 '13 at 18:46

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