Diophantine approximation with sums of two squares II
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912869782650880 |
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| author | Baier, Stephan Rahaman, Habibur |
| author_facet | Baier, Stephan Rahaman, Habibur |
| contents | Recently, the authors showed that for every irrational number $α$, there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$, satisfying $||αn||<n^{-(1/2-\varepsilon)}$ for any fixed $\varepsilon>0$. We also provided a quantitative version with a lower bound when the exponent $1/2-\varepsilon$ is replaced by a smaller exponent $γ<3/7-\varepsilon$. In this article, we establish a quantitative version for the exponent $1/2-\varepsilon$, where we confine ourselves to the particular case of sums of two squares. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_18044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diophantine approximation with sums of two squares II Baier, Stephan Rahaman, Habibur Number Theory 11J25, 11J54, 11J71, 11L05, 11E25 Recently, the authors showed that for every irrational number $α$, there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$, satisfying $||αn||<n^{-(1/2-\varepsilon)}$ for any fixed $\varepsilon>0$. We also provided a quantitative version with a lower bound when the exponent $1/2-\varepsilon$ is replaced by a smaller exponent $γ<3/7-\varepsilon$. In this article, we establish a quantitative version for the exponent $1/2-\varepsilon$, where we confine ourselves to the particular case of sums of two squares. |
| title | Diophantine approximation with sums of two squares II |
| topic | Number Theory 11J25, 11J54, 11J71, 11L05, 11E25 |
| url | https://arxiv.org/abs/2508.18044 |