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arXiv · 0709.3419

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

Nikolai G. Moshchevitin
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Abstract

Let the sequence $\{t_n\}_{n=1}^{\infty}$ of reals satisfy the condition $ \frac{t_{n+1}}{t_n} \ge 1+ \fracγ{n^β},0\le β<1, γ>0. $ Then the set $ \{α\in [0,1]: \exists \varkappa > 0 \forall n \in \mathbb{N} ||t_n α|| > \frac{\varkappa}{n^β\log (n+1)} \} $ is uncountable. Moreover its Hausdorff dimension is equal to 1. Consider the set of naturals of the form $2^n3^m$ 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 $ \{α\in [0,1]: \exists \varkappa > 0 \forall n \in \mathbb{N} ||s_n α|| > \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.

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Nikolai G. Moshchevitin. 2007-10-20. Density modulo 1 of sublacunary sequences: application of Peres-Schlag's arguments. https://arxiv.org/abs/0709.3419

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