Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913973409939456 |
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| author | Neckrasov, Vasiliy |
| author_facet | Neckrasov, Vasiliy |
| contents | In [Compositio Math. 155 (2019)] Kleinbock and Wadleigh proved a "zero-one law" for uniform inhomogeneous Diophantine approximations. We generalize this statement with arbitrary weight functions and establish a new and simple proof of this statement, based on transference principle. We also give a complete description of the sets of $g$-Dirichlet pairs with a fixed matrix in this setup from Lebesgue measure point of view. As an application, we consider the set of badly approximable matrices and give a characterization of bad approximability in terms of inhomogeneous approximations. All the aforementioned metrical descriptions work (and sometimes can be strengthened) for weighted Diophantine approximations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_01912 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle Neckrasov, Vasiliy Number Theory Dynamical Systems 11J20 (Primary) 11J83, 11J13, 37A44 (Secondary) In [Compositio Math. 155 (2019)] Kleinbock and Wadleigh proved a "zero-one law" for uniform inhomogeneous Diophantine approximations. We generalize this statement with arbitrary weight functions and establish a new and simple proof of this statement, based on transference principle. We also give a complete description of the sets of $g$-Dirichlet pairs with a fixed matrix in this setup from Lebesgue measure point of view. As an application, we consider the set of badly approximable matrices and give a characterization of bad approximability in terms of inhomogeneous approximations. All the aforementioned metrical descriptions work (and sometimes can be strengthened) for weighted Diophantine approximations. |
| title | Khintchine-type theorems for weighted uniform inhomogeneous approximations via transference principle |
| topic | Number Theory Dynamical Systems 11J20 (Primary) 11J83, 11J13, 37A44 (Secondary) |
| url | https://arxiv.org/abs/2508.01912 |