Diophantine approximation with sums of two squares
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908932188930048 |
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| author | Baier, Stephan Rahaman, Habibur |
| author_facet | Baier, Stephan Rahaman, Habibur |
| contents | For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $α$, there exist infinitely many positive integers $n$ represented by $Q$ and satisfying $||αn||<n^{-(1/2-\varepsilon)}$ for any fixed but arbitrarily small $\varepsilon>0$. This is an easy consequence of a result by Cook on small fractional parts of diagonal quadratic forms. Secondly, we give a quantitative version with a lower bound of this result when the exponent $1/2-\varepsilon$ is replaced by any fixed $γ<3/7$. To this end, we use the Voronoi summation formula and a bound for bilinear forms with Kloosterman sums to fixed moduli by Kerr, Shparlinski, Wu and Xi. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09650 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diophantine approximation with sums of two squares Baier, Stephan Rahaman, Habibur Number Theory 11J25, 11J54, 11J71, 11L05, 11E25 For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $α$, there exist infinitely many positive integers $n$ represented by $Q$ and satisfying $||αn||<n^{-(1/2-\varepsilon)}$ for any fixed but arbitrarily small $\varepsilon>0$. This is an easy consequence of a result by Cook on small fractional parts of diagonal quadratic forms. Secondly, we give a quantitative version with a lower bound of this result when the exponent $1/2-\varepsilon$ is replaced by any fixed $γ<3/7$. To this end, we use the Voronoi summation formula and a bound for bilinear forms with Kloosterman sums to fixed moduli by Kerr, Shparlinski, Wu and Xi. |
| title | Diophantine approximation with sums of two squares |
| topic | Number Theory 11J25, 11J54, 11J71, 11L05, 11E25 |
| url | https://arxiv.org/abs/2504.09650 |