Uniform Diophantine approximation with restricted denominators
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909134248476672 |
|---|---|
| author | Wang, Bo Li, Bing Li, Ruofan |
| author_facet | Wang, Bo Li, Bing Li, Ruofan |
| contents | Let $b\geq2$ be an integer and $A=(a_{n})_{n=1}^{\infty}$ be a strictly increasing subsequence of positive integers with $η:=\limsup\limits_{n\to\infty}\frac{a_{n+1}}{a_{n}}<+\infty$. For each irrational real number $ξ$, we denote by $\hat{v}_{b,A}(ξ)$ the supremum of the real numbers $\hat{v}$ for which, for every sufficiently large integer $N$, the equation $\|b^{a_n}ξ\|<(b^{a_N})^{-\hat{v}}$ has a solution $n$ with $1\leq n\leq N$. For every $\hat{v}\in[0,η]$, let $\hat{\mathcal{V}}_{b,A}(\hat{v})$ ($\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$) be the set of all real numbers $ξ$ such that $\hat{v}_{b,A}(ξ)\geq\hat{v}$ ($\hat{v}_{b,A}(ξ)=\hat{v}$) respectively. In this paper, we give some results of the Hausdorfff dimensions of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ and $\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$. When $η=1$, we prove that the Hausdorfff dimensions of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ and $\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$ are equal to $\left(\frac{1-\hat{v}}{1+\hat{v}}\right)^{2}$ for any $\hat{v}\in[0,1]$. When $η>1$ and $\lim_{n\to\infty}\frac{a_{n+1}}{a_{n}}$ exists, we show that the Hausdorfff dimension of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ is strictly less than $\left(\frac{η-\hat{v}}{η+\hat{v}}\right)^{2}$ for some $\hat{v}$, which is different with the case $η=1$, and we give a lower bound of the Hausdorfff dimensions of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ and $\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$ for any $\hat{v}\in[0,η]$. Furthermore, we show that this lower bound can be reached for some $\hat{v}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_03923 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniform Diophantine approximation with restricted denominators Wang, Bo Li, Bing Li, Ruofan Number Theory Dynamical Systems Let $b\geq2$ be an integer and $A=(a_{n})_{n=1}^{\infty}$ be a strictly increasing subsequence of positive integers with $η:=\limsup\limits_{n\to\infty}\frac{a_{n+1}}{a_{n}}<+\infty$. For each irrational real number $ξ$, we denote by $\hat{v}_{b,A}(ξ)$ the supremum of the real numbers $\hat{v}$ for which, for every sufficiently large integer $N$, the equation $\|b^{a_n}ξ\|<(b^{a_N})^{-\hat{v}}$ has a solution $n$ with $1\leq n\leq N$. For every $\hat{v}\in[0,η]$, let $\hat{\mathcal{V}}_{b,A}(\hat{v})$ ($\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$) be the set of all real numbers $ξ$ such that $\hat{v}_{b,A}(ξ)\geq\hat{v}$ ($\hat{v}_{b,A}(ξ)=\hat{v}$) respectively. In this paper, we give some results of the Hausdorfff dimensions of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ and $\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$. When $η=1$, we prove that the Hausdorfff dimensions of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ and $\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$ are equal to $\left(\frac{1-\hat{v}}{1+\hat{v}}\right)^{2}$ for any $\hat{v}\in[0,1]$. When $η>1$ and $\lim_{n\to\infty}\frac{a_{n+1}}{a_{n}}$ exists, we show that the Hausdorfff dimension of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ is strictly less than $\left(\frac{η-\hat{v}}{η+\hat{v}}\right)^{2}$ for some $\hat{v}$, which is different with the case $η=1$, and we give a lower bound of the Hausdorfff dimensions of $\hat{\mathcal{V}}_{b,A}(\hat{v})$ and $\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$ for any $\hat{v}\in[0,η]$. Furthermore, we show that this lower bound can be reached for some $\hat{v}$. |
| title | Uniform Diophantine approximation with restricted denominators |
| topic | Number Theory Dynamical Systems |
| url | https://arxiv.org/abs/2302.03923 |