Uniform Diophantine approximation with restricted denominators

TL;DR

This paper investigates the Hausdorff dimensions of sets of real numbers with specific Diophantine approximation properties involving restricted denominators, providing explicit formulas and bounds depending on the growth rate of the subsequence.

Contribution

It establishes explicit Hausdorff dimension formulas for approximation sets with restricted denominators, especially distinguishing cases where the subsequence growth rate is 1 or greater than 1.

Findings

Dimension equals rac{(1- ext{v})^2}{(1+ ext{v})^2} when ext{eta}=1.

Dimension is strictly less than rac{( ext{eta}- ext{v})^2}{( ext{eta}+ ext{v})^2} for some ext{v} when ext{eta}>1.

Lower bounds of dimensions are provided and are attainable for certain ext{v}.

Abstract

Let $b\geq2$ be an integer and $A=(a_{n})_{n=1}^{\infty}$ be a strictly increasing subsequence of positive integers with $\eta:=\limsup\limits_{n\to\infty}\frac{a_{n+1}}{a_{n}}<+\infty$. For each irrational real number $\xi$, we denote by $\hat{v}_{b,A}(\xi)$ the supremum of the real numbers $\hat{v}$ for which, for every sufficiently large integer $N$, the equation $\|b^{a_n}\xi\|<(b^{a_N})^{-\hat{v}}$ has a solution $n$ with $1\leq n\leq N$. For every $\hat{v}\in[0,\eta]$, let $\hat{\mathcal{V}}_{b,A}(\hat{v})$ ($\hat{\mathcal{V}}_{b,A}^{\ast}(\hat{v})$) be the set of all real numbers $\xi$ such that $\hat{v}_{b,A}(\xi)\geq\hat{v}$ ($\hat{v}_{b,A}(\xi)=\hat{v}$) 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 $\eta=1$, 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 $\eta>1$ 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 $\left(\frac{\eta-\hat{v}}{\eta+\hat{v}}\right)^{2}$ for some $\hat{v}$, which is different with the case $\eta=1$, 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 $\hat{v}\in[0,\eta]$. Furthermore, we show that this lower bound can be reached for some $\hat{v}$.