Uniform Diophantine approximation with restrictions via total density of collections of subspaces
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| Format: | Preprint |
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2026
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| _version_ | 1866912984377327616 |
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| author | Hong, Leo Kleinbock, Dmitry Neckrasov, Vasiliy |
| author_facet | Hong, Leo Kleinbock, Dmitry Neckrasov, Vasiliy |
| contents | In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of $n \geq 2$ variables. Throughout the years, this argument has been extensively modified and generalized. Most recently, Kleinbock et al. (2025) introduced a general framework of Diophantine systems and showed that a certain topological property called total density implies a far-reaching generalization of Khintchine's result. We describe a way to establish total density for a variety of Diophantine systems, and thus prove that the sets of singular objects are uncountable and dense in a wide range of set-ups in Diophantine approximation. As a special case, we establish such a result for inhomogeneous approximation, proving the existence of uncountably many singular systems of affine forms with a fixed translation part. One can also consider approximation with prime denominators, or more generally, approximation under some strong restrictions on numerators and denominators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_25988 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform Diophantine approximation with restrictions via total density of collections of subspaces Hong, Leo Kleinbock, Dmitry Neckrasov, Vasiliy Number Theory 11J13, 11J20, 14N20 In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of $n \geq 2$ variables. Throughout the years, this argument has been extensively modified and generalized. Most recently, Kleinbock et al. (2025) introduced a general framework of Diophantine systems and showed that a certain topological property called total density implies a far-reaching generalization of Khintchine's result. We describe a way to establish total density for a variety of Diophantine systems, and thus prove that the sets of singular objects are uncountable and dense in a wide range of set-ups in Diophantine approximation. As a special case, we establish such a result for inhomogeneous approximation, proving the existence of uncountably many singular systems of affine forms with a fixed translation part. One can also consider approximation with prime denominators, or more generally, approximation under some strong restrictions on numerators and denominators. |
| title | Uniform Diophantine approximation with restrictions via total density of collections of subspaces |
| topic | Number Theory 11J13, 11J20, 14N20 |
| url | https://arxiv.org/abs/2603.25988 |