On the fractional parts of certain sequences of $ξα^{n}$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916039890042880 |
|---|---|
| author | Gao, Xiang Yip, Chi Hoi |
| author_facet | Gao, Xiang Yip, Chi Hoi |
| contents | Assume that $α>1$ is an algebraic number and $ξ\neq0$ is a real number. We are concerned with the distribution of the fractional parts of the sequence $(ξα^{n})$. Under various Diophantine conditions on $ξ$ and $α$, we obtain lower bounds on the number $n$ with $1\leq n\leq N $ for which the fractional part of the sequence $(ξα^{n})_{n\geq1}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02972 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the fractional parts of certain sequences of $ξα^{n}$ Gao, Xiang Yip, Chi Hoi Number Theory Classical Analysis and ODEs Primary: 11J71, 28A80. Secondary: 11K16, 37A45, 42A38 Assume that $α>1$ is an algebraic number and $ξ\neq0$ is a real number. We are concerned with the distribution of the fractional parts of the sequence $(ξα^{n})$. Under various Diophantine conditions on $ξ$ and $α$, we obtain lower bounds on the number $n$ with $1\leq n\leq N $ for which the fractional part of the sequence $(ξα^{n})_{n\geq1}$ 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. |
| title | On the fractional parts of certain sequences of $ξα^{n}$ |
| topic | Number Theory Classical Analysis and ODEs Primary: 11J71, 28A80. Secondary: 11K16, 37A45, 42A38 |
| url | https://arxiv.org/abs/2408.02972 |