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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2401.08849 |
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| _version_ | 1866914643130187776 |
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| author | Pavlenkov, Volodymyr Zorin, Evgeniy |
| author_facet | Pavlenkov, Volodymyr Zorin, Evgeniy |
| contents | We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence of denominators and the decay rate of the Fourier transform of a Rajchman measure. Among the other things, this allows applications to sequences of denominators of polynomial growth. In particular, we infer new inhomogeneous Khintchine-Järnik type theorems with restraint denominators for a broad family of denominator sequences. Furthermore, our results provide non-trivial lower bounds for Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers with restraint denominators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_08849 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inhomogeneous Diophantine Approximation on $M_0$-sets Pavlenkov, Volodymyr Zorin, Evgeniy Number Theory Dynamical Systems 11J83 (Primary), 42A61 (Secondary) We prove new quantitative Schmidt-type theorem for Diophantine approximations with restraint denominators on fractals (more precisely, on $M_0$-sets). Our theorems introduce a sharp balance condition between the growth rate of the sequence of denominators and the decay rate of the Fourier transform of a Rajchman measure. Among the other things, this allows applications to sequences of denominators of polynomial growth. In particular, we infer new inhomogeneous Khintchine-Järnik type theorems with restraint denominators for a broad family of denominator sequences. Furthermore, our results provide non-trivial lower bounds for Hausdorff dimensions of intersections of two sets of inhomogeneously well-approximable numbers with restraint denominators. |
| title | Inhomogeneous Diophantine Approximation on $M_0$-sets |
| topic | Number Theory Dynamical Systems 11J83 (Primary), 42A61 (Secondary) |
| url | https://arxiv.org/abs/2401.08849 |