Simultaneous Khintchine theorem on manifolds in positive characteristics: convergence case
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917304610062336 |
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| author | Aranov, Noy Soffer Das, Sourav Ganguly, Arijit Pandey, Aratrika |
| author_facet | Aranov, Noy Soffer Das, Sourav Ganguly, Arijit Pandey, Aratrika |
| contents | We prove the convergence case of Khintchine's theorem, with general approximation functions that are not necessarily monotonic, for analytic nonplanar manifolds over local fields of positive characteristic. Our approach is based on the method of counting rational points near manifolds developed by Beresnevich and Yang. To address the scenario where the given approximating function is not monotonic, we extend our function field by adjoining an appropriate root. Additionally, in the course of the proof, we establish several new results in the geometry of numbers over function fields, which are of independent interest. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_01991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simultaneous Khintchine theorem on manifolds in positive characteristics: convergence case Aranov, Noy Soffer Das, Sourav Ganguly, Arijit Pandey, Aratrika Number Theory Dynamical Systems 11K60 We prove the convergence case of Khintchine's theorem, with general approximation functions that are not necessarily monotonic, for analytic nonplanar manifolds over local fields of positive characteristic. Our approach is based on the method of counting rational points near manifolds developed by Beresnevich and Yang. To address the scenario where the given approximating function is not monotonic, we extend our function field by adjoining an appropriate root. Additionally, in the course of the proof, we establish several new results in the geometry of numbers over function fields, which are of independent interest. |
| title | Simultaneous Khintchine theorem on manifolds in positive characteristics: convergence case |
| topic | Number Theory Dynamical Systems 11K60 |
| url | https://arxiv.org/abs/2511.01991 |