Weak approximations, Diophantine exponents and two-dimensional lattices
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912983022567424 |
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| author | Moshchevitin, Nikolay |
| author_facet | Moshchevitin, Nikolay |
| contents | We study properties of Diophantine exponents of lattices and so-called related "weak" uniform approximations introduced in recent papers by Oleg German, in the simplest two-dimensional case. In contrast to the multidimensional case, in the two-dimensional case we can use a powerful tool of continued fractions. We develop an analog of Jarn\'ık's theory dealing with inequalities between the ordinary and uniform Diophantine exponents, which turned out to be related to mutual behaviour of irrationality measure functions for two real numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_25071 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak approximations, Diophantine exponents and two-dimensional lattices Moshchevitin, Nikolay Number Theory 11J04, 11J70, 11J13 We study properties of Diophantine exponents of lattices and so-called related "weak" uniform approximations introduced in recent papers by Oleg German, in the simplest two-dimensional case. In contrast to the multidimensional case, in the two-dimensional case we can use a powerful tool of continued fractions. We develop an analog of Jarn\'ık's theory dealing with inequalities between the ordinary and uniform Diophantine exponents, which turned out to be related to mutual behaviour of irrationality measure functions for two real numbers. |
| title | Weak approximations, Diophantine exponents and two-dimensional lattices |
| topic | Number Theory 11J04, 11J70, 11J13 |
| url | https://arxiv.org/abs/2603.25071 |