Diophantine approximation with primes from short intervals
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908938312613888 |
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| author | Baier, Stephan Roy, Sayantan |
| author_facet | Baier, Stephan Roy, Sayantan |
| contents | In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If $α$ is an irrational number having a continued fraction expansion with bounded terms (in particular, if $α$ is a quadratic irrational), then the number of primes $p$ in the interval $(X-Y,X]$ satisfying $||pα||<δ$ is asymptotically equal to $2δY/\log X$, provided that $X\ge 10$, $X^{2/3+\varepsilon}\le Y\le X/2$ and $X^{\varepsilon}\max\left\{X^{1/4}Y^{-1/2},X^{2/3}Y^{-1}\right\}\le δ\le 1/2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_02174 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diophantine approximation with primes from short intervals Baier, Stephan Roy, Sayantan Number Theory 11J71, 11L20, 11N05, 11N35 In this paper, we establish hybrid results on Diophantine approximation with primes from short intervals. In particular, we prove the following result in a slightly modified form: If $α$ is an irrational number having a continued fraction expansion with bounded terms (in particular, if $α$ is a quadratic irrational), then the number of primes $p$ in the interval $(X-Y,X]$ satisfying $||pα||<δ$ is asymptotically equal to $2δY/\log X$, provided that $X\ge 10$, $X^{2/3+\varepsilon}\le Y\le X/2$ and $X^{\varepsilon}\max\left\{X^{1/4}Y^{-1/2},X^{2/3}Y^{-1}\right\}\le δ\le 1/2$. |
| title | Diophantine approximation with primes from short intervals |
| topic | Number Theory 11J71, 11L20, 11N05, 11N35 |
| url | https://arxiv.org/abs/2512.02174 |