Squarefree numbers in short intervals
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909134925856768 |
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| author | Pandey, Mayank |
| author_facet | Pandey, Mayank |
| contents | We show that there exists $η> 0$ such that the interval $[X, X + X^{\frac 15 - η}]$ contains a squarefree number for all large $X$. This improves on an earlier result of Filaseta and Trifonov who showed that there is a squarefree number in $[X, X + cX^{\frac 15}\log X]$ for some $c > 0$ and all large $X$.
We introduce a new technique to count lattice points near curves, which we use to bound in critical ranges the number of integers in a short interval divisible by a large square. This uses as an input Green and Tao's quantitative version of Leibman's theorem on the equidistribution of polynomial orbits in nilmanifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13981 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Squarefree numbers in short intervals Pandey, Mayank Number Theory 11N25 We show that there exists $η> 0$ such that the interval $[X, X + X^{\frac 15 - η}]$ contains a squarefree number for all large $X$. This improves on an earlier result of Filaseta and Trifonov who showed that there is a squarefree number in $[X, X + cX^{\frac 15}\log X]$ for some $c > 0$ and all large $X$. We introduce a new technique to count lattice points near curves, which we use to bound in critical ranges the number of integers in a short interval divisible by a large square. This uses as an input Green and Tao's quantitative version of Leibman's theorem on the equidistribution of polynomial orbits in nilmanifolds. |
| title | Squarefree numbers in short intervals |
| topic | Number Theory 11N25 |
| url | https://arxiv.org/abs/2401.13981 |