Density modulo 1 of sublacunary sequences: application of Peres-Schlag's arguments

Abstract

Let the sequence $\{t_n{\}}_{n=1}^{\infty }$ of reals satisfy the condition $\frac{t_{n+1}}{t_n}\ge 1+\frac{\gamma }{n^{\beta }},0\le \beta <1,\gamma >0.$ Then the set $\{\alpha \in [0,1]:\exists \varkappa >0\forall n\in \mathbb{N}||t_n\alpha ||>\frac{\varkappa }{n^{\beta }\log (n+1)}\}$ is uncountable. Moreover its Hausdorff dimension is equal to 1. Consider the set of naturals of the form $2^n{3}^m$ and let the sequence $s_1=1,s_2=2,s_3=3,s_4=4,s_5=6,s_6=8,...$ performs this set as an increasing sequence. Then the set $\{\alpha \in [0,1]:\exists \varkappa >0\forall n\in \mathbb{N}||s_n\alpha ||>\frac{\varkappa }{\sqrt{n}\log (n+1)}\}$ also has Hausdorff dimension equal to 1. The results obtained use an original approach due to Y. Peres and W. Schlag.

9 pages, minor correction in Section 6B

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