Ratio of sum of digits functions in two bases

In 2019 La Bretèche, Stoll and Tennenbaum showed that the ratio of the sum of digits function $s_{q_1}(n)\mathrm{/}{s}_{q_2}(n)$ of two multiplicatively independent bases $q_1$ and $q_2$ is dense in ${\mathbb{Q}}^{+}$. Recently Spiegelhofer proved that in the special case $s_2(n)\mathrm{/}{s}_3(n)=1$ we have infinitely many solutions. Spiegelhofer extended this jointly with Drmota to show that the pair $(s_2(n),s_3(n))$ attains almost every value of ${\mathbb{N}}^2$ and hence, in particular, that every rational ratio is attained infinitely many times.\ In this paper we show that, indeed, for any pair of multiplicatively independent bases $p$ and $q$, that the ratio attains every rational number infinitely many times. We also study this problem in the multiplicatively dependent case, hence giving a complete characterisation in the case of 2 bases.