Geodesic
Density Modulo 1 of Sublacunary Sequences
Matematičeskie zametki, Tome 77 (2005) no. 6, pp. 803-813.

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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. 
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R. K. Akhunzhanov; N. G. Moshchevitin. Density Modulo 1 of Sublacunary Sequences. Matematičeskie zametki, Tome 77 (2005) no. 6, pp. 803-813. http://geodesic.mathdoc.fr/item/MZM_2005_77_6_a0/

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