Lower bounds for the Hausdorff dimension of expressible sets

We obtain positive lower bounds on the Hausdorff dimension of sets of real numbers given by expressions of the form $\sum_{n=1}^\infty \frac{1}{a_n b_n}$, where $b_n$ satisfies some growth condition and $a_n$ lies in some set, possibly depending on $n$. As a consequence of our results, some of the irrational numbers arising from Erd\H{o}s' celebrated construction from 1976 are not Liouville numbers.