A note on general isolation result in Diophantine Approximation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912649553379328 |
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| author | Pitcyn, Sergei Moshchevitin, Nikolay |
| author_facet | Pitcyn, Sergei Moshchevitin, Nikolay |
| contents | In the present paper we give very simple general statements which deal with approximation of a real number by rationals and are related to isolation phenomenon. In particular we study functions $ f(x)>f_1(x)>0$ such that existence of solutions $\frac{p}{q}$ of Diophantine inequality $ \left| α-\frac{p}{q}\right|< \frac{f(q)}{q^2} $ leads to the existence of solutions of inequality $ \left| α-\frac{p}{q}\right|< \frac{f_1(q)}{q^2} $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25628 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on general isolation result in Diophantine Approximation Pitcyn, Sergei Moshchevitin, Nikolay Number Theory 11J04, 11J06, 11J13 In the present paper we give very simple general statements which deal with approximation of a real number by rationals and are related to isolation phenomenon. In particular we study functions $ f(x)>f_1(x)>0$ such that existence of solutions $\frac{p}{q}$ of Diophantine inequality $ \left| α-\frac{p}{q}\right|< \frac{f(q)}{q^2} $ leads to the existence of solutions of inequality $ \left| α-\frac{p}{q}\right|< \frac{f_1(q)}{q^2} $. |
| title | A note on general isolation result in Diophantine Approximation |
| topic | Number Theory 11J04, 11J06, 11J13 |
| url | https://arxiv.org/abs/2509.25628 |