Diophantine approximation with integers having no large prime factors
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914431885115392 |
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| author | Nath, Kunjakanan Rahaman, Habibur |
| author_facet | Nath, Kunjakanan Rahaman, Habibur |
| contents | Given any irrational number $α$, we show that for any $0<θ<6/17$, there are infinitely many $y$-smooth (friable) numbers $n$ such that
$$\|nα\| < n^{-θ},$$ where $(\log n)^C\leq y\leq n$ for some large constant $C>0$. This improves the previous work of Baker, who obtained the exponent $1/3-2/(3C)+o(1)$ in the case of $y\geq (\log n)^C$, and that of Yau, who obtained the exponent $1/3$ when $y=n^{o(1)}$. Our proof is based on the dispersion method together with arithmetic inputs coming from the average bounds for Kloosterman sums over smooth numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17732 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Diophantine approximation with integers having no large prime factors Nath, Kunjakanan Rahaman, Habibur Number Theory 11J25, 11J71, 11L05, 11L07 Given any irrational number $α$, we show that for any $0<θ<6/17$, there are infinitely many $y$-smooth (friable) numbers $n$ such that $$\|nα\| < n^{-θ},$$ where $(\log n)^C\leq y\leq n$ for some large constant $C>0$. This improves the previous work of Baker, who obtained the exponent $1/3-2/(3C)+o(1)$ in the case of $y\geq (\log n)^C$, and that of Yau, who obtained the exponent $1/3$ when $y=n^{o(1)}$. Our proof is based on the dispersion method together with arithmetic inputs coming from the average bounds for Kloosterman sums over smooth numbers. |
| title | Diophantine approximation with integers having no large prime factors |
| topic | Number Theory 11J25, 11J71, 11L05, 11L07 |
| url | https://arxiv.org/abs/2603.17732 |