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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.10863 |
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| _version_ | 1866908537844662272 |
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| author | Fu, Xiaohui Shi, Junjie Tian, Chen |
| author_facet | Fu, Xiaohui Shi, Junjie Tian, Chen |
| contents | In this paper, we investigate the two-dimensional uniform Diophantine approximation in $β$-dynamical systems. Let $β_i > 1(i=1,2)$ be real numbers, and let $T_{β_i}$ denote the $β_i$-transformation defined on $[0, 1]$. For each $(x, y) \in[0,1]^2$, we define the asymptotic approximation exponent $$ v_{β_1, β_2}(x, y)=\sup \left\{0 \leq v<\infty: \begin{array}{l} T_{β_1}^n x<β_1^{-n v} \\ T_{β_2}^n y<β_2^{-n v} \end{array} \text { for infinitely many } n \in \mathbb{N}\right\} \text {, } $$ and the uniform approximation exponent $$ \hat{v}_{β_1, β_2}(x, y)=\sup \left\{0 \leq \hat{v}<\infty: \forall~ N \gg 1, \exists 1 \leq n \leq N \text { such that } \begin{array}{l} T_{β_1}^n x < β_1^{-N \hat{v}} \\ T_{β_2}^n y < β_2^{-N \hat{v}} \end{array}\right\} . $$ We calculate the Hausdorff dimension of the intersection $$\left\{(x, y) \in[0,1]^2: \hat{v}_{β_1, β_2}(x, y)=\hat{v} \text { and } v_{β_1, β_2}(x, y)=v\right\}$$ for any $\hat{v}$ and $v$ satisfying $\log _{β_2}{β_1}>\frac{\hat{v}}{v}(1+v)$. As a corollary, we establish a definite formula for the Hausdorff dimension of the level set of the uniform approximation exponent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_10863 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform Diophantine approximation on the plane for $β$-dynamical systems Fu, Xiaohui Shi, Junjie Tian, Chen Dynamical Systems In this paper, we investigate the two-dimensional uniform Diophantine approximation in $β$-dynamical systems. Let $β_i > 1(i=1,2)$ be real numbers, and let $T_{β_i}$ denote the $β_i$-transformation defined on $[0, 1]$. For each $(x, y) \in[0,1]^2$, we define the asymptotic approximation exponent $$ v_{β_1, β_2}(x, y)=\sup \left\{0 \leq v<\infty: \begin{array}{l} T_{β_1}^n x<β_1^{-n v} \\ T_{β_2}^n y<β_2^{-n v} \end{array} \text { for infinitely many } n \in \mathbb{N}\right\} \text {, } $$ and the uniform approximation exponent $$ \hat{v}_{β_1, β_2}(x, y)=\sup \left\{0 \leq \hat{v}<\infty: \forall~ N \gg 1, \exists 1 \leq n \leq N \text { such that } \begin{array}{l} T_{β_1}^n x < β_1^{-N \hat{v}} \\ T_{β_2}^n y < β_2^{-N \hat{v}} \end{array}\right\} . $$ We calculate the Hausdorff dimension of the intersection $$\left\{(x, y) \in[0,1]^2: \hat{v}_{β_1, β_2}(x, y)=\hat{v} \text { and } v_{β_1, β_2}(x, y)=v\right\}$$ for any $\hat{v}$ and $v$ satisfying $\log _{β_2}{β_1}>\frac{\hat{v}}{v}(1+v)$. As a corollary, we establish a definite formula for the Hausdorff dimension of the level set of the uniform approximation exponent. |
| title | Uniform Diophantine approximation on the plane for $β$-dynamical systems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2509.10863 |