On the distribution of $αp^2$ modulo one in the intersection of two Piatetski--Shapiro sets
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913081265750016 |
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| author | Chu, Junyi Li, Jinjiang Zhang, Min |
| author_facet | Chu, Junyi Li, Jinjiang Zhang, Min |
| contents | Let $\lfloor t\rfloor$ denote the integer part of $t\in\mathbb{R}$ and $\|x\|$ the distance from $x$ to the nearest integer. Suppose that $1/2<γ_2<γ_1<1$ are two fixed constants. In this paper, it is proved that, whenever $α$ is an irrational number and $β$ is any real number, there exist infinitely many prime numbers $p$ in the intersection of two Piatetski--Shapiro sets, i.e., $p=\lfloor n_1^{1/γ_1}\rfloor=\lfloor n_2^{1/γ_2}\rfloor$, such that \begin{equation*} \|αp^2+β\|<p^{-\frac{14(γ_1+γ_2)-27}{43}+\varepsilon}, \end{equation*} provided that $27/14<γ_1+γ_2<2$. This result constitutes an generalization upon the previous result of Dimitrov [4]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02775 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the distribution of $αp^2$ modulo one in the intersection of two Piatetski--Shapiro sets Chu, Junyi Li, Jinjiang Zhang, Min Number Theory Let $\lfloor t\rfloor$ denote the integer part of $t\in\mathbb{R}$ and $\|x\|$ the distance from $x$ to the nearest integer. Suppose that $1/2<γ_2<γ_1<1$ are two fixed constants. In this paper, it is proved that, whenever $α$ is an irrational number and $β$ is any real number, there exist infinitely many prime numbers $p$ in the intersection of two Piatetski--Shapiro sets, i.e., $p=\lfloor n_1^{1/γ_1}\rfloor=\lfloor n_2^{1/γ_2}\rfloor$, such that \begin{equation*} \|αp^2+β\|<p^{-\frac{14(γ_1+γ_2)-27}{43}+\varepsilon}, \end{equation*} provided that $27/14<γ_1+γ_2<2$. This result constitutes an generalization upon the previous result of Dimitrov [4]. |
| title | On the distribution of $αp^2$ modulo one in the intersection of two Piatetski--Shapiro sets |
| topic | Number Theory |
| url | https://arxiv.org/abs/2312.02775 |