Abstract:Assume that $\alpha>1$ is an algebraic number and $\xi\neq0$ is a real number. We are concerned with the distribution of the fractional parts of the sequence $(\xi \alpha^{n})$. Under various Diophantine conditions on $\xi$ and $\alpha$, we obtain lower bounds on the number $n$ with $1\leq n\leq N $ for which the fractional part of the sequence $(\xi \alpha^{n})_{n\geq1}$ fall into a prescribed region $I\subset [0,1]$, extending several results in the literature. As an application, we show that the Fourier decay rate of some self-similar measures is logarithmic, generalizing a result of Varjú and Yu.
| Comments: | 13 pages, minor correction |
| Subjects: | Number Theory (math.NT); Classical Analysis and ODEs (math.CA) |
| MSC classes: | Primary: 11J71, 28A80. Secondary: 11K16, 37A45, 42A38 |
| Cite as: | arXiv:2408.02972 [math.NT] |
| (or arXiv:2408.02972v2 [math.NT] for this version) | |
| https://doi.org/10.48550/arXiv.2408.02972 arXiv-issued DOI via DataCite |
From: Chi Hoi Yip [view email]
[v1]
Tue, 6 Aug 2024 06:06:23 UTC (13 KB)
[v2]
Sat, 23 May 2026 04:10:45 UTC (15 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。