Density modulo 1 of sublacunary sequences (Q2388189)

Let \((t_n)_{n=1}^{\infty}\) be a sequence of real numbers such that \(t_{n+1}/t_n\geq 1+c/n^{\beta}\) with \(0\leq \beta \leq 1/2\), \(c>0\) (such sequences are said to be sublacunar). The authors show that there exists a set \(\mathcal{A}\) of real numbers whose Hausdorff dimension is \(1\) such that for each \(\alpha \in \mathcal{A}\) with some positive constant \(\gamma = \gamma(\alpha)\), \(\| t_{n}\alpha\| \geq \gamma/n^{2\beta}\) holds for any natural number \(n\). Consider the ascending sequence \((s_n)_{n=1}^{\infty}\) generated by the ordered numbers of the form \(2^i3^j\), \(i, j=0,1,2,3,\ldots\). Let \(\gamma \leq \gamma_0=2^{-14}\). The authors prove that there exists a set \(\mathcal{A}_{\gamma}\) such that the Hausdorff dimension of \(\mathcal{A_{\gamma}}\) is greater than or equal to \(1-4/(\log_2(1/\gamma)-10)\) and for any \(\alpha \in \mathcal{A}_{\gamma}\), \(\| s_n\alpha \| \geq \gamma/n\) holds for any natural number \(n\). It is shown that the Hausdorff dimension of the set \(\cup_{\gamma>0}\mathcal{A}_{\gamma}\) is equal to \(1\) and the Lebesgue measure of the set is zero.