Lower bounds for the Hausdorff dimension of expressible sets

Abstract

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 Erdős' celebrated construction from 1976 are not Liouville numbers.

9 pages, 2 figures