Density modulo 1 of dilations of sublacunary sequences
\textit{B. de Mathan} and \textit{A. Pollington} independently proved the following result (answering a question by P. Erdős): Let \(\{r_ i\}\) be a lacunary sequence, then the Hausdorff dimension of the set \(E = \{\alpha \in \mathbb{R} : \{r_ i\}\) is not dense mod 1\} is 1 [cf. Acta Math. Acad. Sci. Hung. 36, 237-241 (1980; Zbl 0465.10040) and Ill. J. Math. 23, 511-515 (1979; Zbl 0413.10042)], respectively. The main result of the present paper is concerned with sequences \(\{r_ i\}\) satisfying \(r_{i+1} / r_ i \to 1\) and \(\limsup r_ i = \infty\). For such sequences it is proved that the Hausdorff dimension of \(E\) is 0. Furthermore for any unbounded \(\{r_ i\}\) with \(r_{i+1} / r_ i < C\) it is shown that \(E\) is a countable union of sets of Hausdorff dimension \( < 1\).
scientific article; zbMATH DE number 706488