Diophantine approximation and the Mass Transference Principle: incorporating the unbounded setup
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914986984472576 |
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| author | Li, Bing Liao, Lingmin Wnag, Baowei Velani, Sanju Zorin, Evgeniy |
| author_facet | Li, Bing Liao, Lingmin Wnag, Baowei Velani, Sanju Zorin, Evgeniy |
| contents | We develop the Mass Transference Principle for rectangles of Wang \& Wu (Math. Ann. 2021) to incorporate the `unbounded' setup; that is, when along some direction the lower order (at infinity) of the side lengths of the rectangles under consideration is infinity. As applications, we obtain the Hausdorff dimension of naturally occurring $\limsup$ sets within the classical framework of simultaneous Diophantine approximation and the dynamical framework of shrinking target problems. For instance, concerning the former, for $τ>0$, let $S(τ)$ denote the set of $(x_1,x_2)\in \mathbb{R}^2$ simultaneously satisfying the inequalities $\|q x_1 \| \, < \, q^{-τ} $ and $ \|q x_2 \| \, < \, e^{-q}$ for infinitely many $q \in \mathbb{N}$. Then, the `unbounded' Mass Transference Principle enables us to show that
$\dim_{\rm H} S(τ) \, = \, \min \big\{ 1, 3/(1+τ) \big\} \, $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_18578 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diophantine approximation and the Mass Transference Principle: incorporating the unbounded setup Li, Bing Liao, Lingmin Wnag, Baowei Velani, Sanju Zorin, Evgeniy Number Theory Dynamical Systems 11J83, 28A78, 28A80, 37E05 We develop the Mass Transference Principle for rectangles of Wang \& Wu (Math. Ann. 2021) to incorporate the `unbounded' setup; that is, when along some direction the lower order (at infinity) of the side lengths of the rectangles under consideration is infinity. As applications, we obtain the Hausdorff dimension of naturally occurring $\limsup$ sets within the classical framework of simultaneous Diophantine approximation and the dynamical framework of shrinking target problems. For instance, concerning the former, for $τ>0$, let $S(τ)$ denote the set of $(x_1,x_2)\in \mathbb{R}^2$ simultaneously satisfying the inequalities $\|q x_1 \| \, < \, q^{-τ} $ and $ \|q x_2 \| \, < \, e^{-q}$ for infinitely many $q \in \mathbb{N}$. Then, the `unbounded' Mass Transference Principle enables us to show that $\dim_{\rm H} S(τ) \, = \, \min \big\{ 1, 3/(1+τ) \big\} \, $. |
| title | Diophantine approximation and the Mass Transference Principle: incorporating the unbounded setup |
| topic | Number Theory Dynamical Systems 11J83, 28A78, 28A80, 37E05 |
| url | https://arxiv.org/abs/2410.18578 |