Simultaneously Small Fractional Parts of Polynomials
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909238156066816 |
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| author | Lau, Cheuk Fung |
| author_facet | Lau, Cheuk Fung |
| contents | Let $f_1,\dots,f_k \in \mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an $n<x$ such that $\|f_i(n)\|\ll x^{-1/10.5kd(d-1)+o(1)}$ for all $1\le i\le k$. This improves on an earlier result of Maynard, who obtained the same exponent dependency on $k$ but not on $d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_01611 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simultaneously Small Fractional Parts of Polynomials Lau, Cheuk Fung Number Theory 11J54, 11J25 Let $f_1,\dots,f_k \in \mathbb{R}[X]$ be polynomials of degree at most $d$ with $f_1(0)=\dots=f_k(0)=0$. We show that there is an $n<x$ such that $\|f_i(n)\|\ll x^{-1/10.5kd(d-1)+o(1)}$ for all $1\le i\le k$. This improves on an earlier result of Maynard, who obtained the same exponent dependency on $k$ but not on $d$. |
| title | Simultaneously Small Fractional Parts of Polynomials |
| topic | Number Theory 11J54, 11J25 |
| url | https://arxiv.org/abs/2407.01611 |