Density Modulo 1 of Sublacunary Sequences

We prove the existence of real numbers badly approximated by rational fractions whose denominators form a sublacunar sequence. For example, for the ascending sequence $s_n$, $n=1,2,3,\dots$, generated by the ordered numbers of the form $2^i3^j$, $i,j=1,2,3,\dots$, we prove that the set of real numbers $\alpha$, such that $\inf_{n\in\mathbb N}n\|s_n\alpha\|>0$, is a set of Hausdorff dimension 1. The divergence of the series $\sum_{n=1}^\infty\frac1n$ implies that the Lebesgue measure of those numbers is zero.