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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2503.10381 |
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| _version_ | 1866916994134048768 |
|---|---|
| author | Xiao, Qian |
| author_facet | Xiao, Qian |
| contents | In one-dimensional Diophantine approximation, the Diophantine properties of a real number are characterized by its partial quotients, especially the growth of its large partial quotients. Notably, Kleinbock and Wadleigh [Proc. Amer. Math. Soc. 2018] made a seminal contribution by linking the improvability of Dirichlet's theorem to the growth of the product of consecutive partial quotients. In this paper, we extend the concept of Dirichlet non-improvable sets within the framework of shrinking target problems. Specifically, consider the dynamical system $([0,1), T)$ of continued fractions. Let $\{z_n\}_{n \ge 1}$ be a sequence of real numbers in $[0,1]$ and let $B > 1$. We determine the Hausdorff dimension of the following set:
\[
\begin{split}
\{x\in[0,1):|T^nx-z_n||T^{n+1}x-Tz_n|<B^{-n}\text{ infinitely often}\}.
\end{split}
\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10381 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Dirichlet non-improvable numbers and shrinking target problems Xiao, Qian Dynamical Systems In one-dimensional Diophantine approximation, the Diophantine properties of a real number are characterized by its partial quotients, especially the growth of its large partial quotients. Notably, Kleinbock and Wadleigh [Proc. Amer. Math. Soc. 2018] made a seminal contribution by linking the improvability of Dirichlet's theorem to the growth of the product of consecutive partial quotients. In this paper, we extend the concept of Dirichlet non-improvable sets within the framework of shrinking target problems. Specifically, consider the dynamical system $([0,1), T)$ of continued fractions. Let $\{z_n\}_{n \ge 1}$ be a sequence of real numbers in $[0,1]$ and let $B > 1$. We determine the Hausdorff dimension of the following set: \[ \begin{split} \{x\in[0,1):|T^nx-z_n||T^{n+1}x-Tz_n|<B^{-n}\text{ infinitely often}\}. \end{split} \] |
| title | On Dirichlet non-improvable numbers and shrinking target problems |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2503.10381 |