论 $$\alpha p$$ modulo one 在两个 Piatetski-Shapiro 集的交集中的分布

Xiaotian Li, Jinjiang Li, Min Zhang
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引用次数: 0

摘要

让\(\lfloor t\rfloor \)表示\(t\in \mathbb {R}\)的整数部分,而\(\Vert x\Vert \)表示从x到最近整数的距离。假设\(1/2<\gamma _2<\gamma _1<1/)是两个固定常数。本文证明,只要 \(α \) 是一个无理数,并且 \(β \) 是任何实数,那么在两个皮亚杰基-沙皮罗集的交集上就存在无穷多个素数 p,即、\p=lfloor n_1^{1/\gamma _1}\rfloor =\lfloor n_2^{1/\gamma _2}\rfloor \),使得$$\begin{aligned}。\α p+beta \Vert <p^{-\frac{12(\gamma _1+\gamma _2)-23}{38}+\varepsilon }, \end{aligned}$$前提是(23/12<\gamma _1+\gamma _2<2)。这一结果是对迪米特洛夫先前结果的概括(《印度纯应用数学》54(3):858-867, 2023)。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the distribution of $$\alpha p$$ modulo one in the intersection of two Piatetski–Shapiro sets

Let \(\lfloor t\rfloor \) denote the integer part of \(t\in \mathbb {R}\) and \(\Vert x\Vert \) the distance from x to the nearest integer. Suppose that \(1/2<\gamma _2<\gamma _1<1\) are two fixed constants. In this paper, it is proved that, whenever \(\alpha \) is an irrational number and \(\beta \) is any real number, there exist infinitely many prime numbers p in the intersection of two Piatetski–Shapiro sets, i.e., \(p=\lfloor n_1^{1/\gamma _1}\rfloor =\lfloor n_2^{1/\gamma _2}\rfloor \), such that

$$\begin{aligned} \Vert \alpha p+\beta \Vert <p^{-\frac{12(\gamma _1+\gamma _2)-23}{38}+\varepsilon }, \end{aligned}$$

provided that \(23/12<\gamma _1+\gamma _2<2\). This result constitutes an generalization upon the previous result of Dimitrov (Indian J Pure Appl Math 54(3):858–867, 2023).

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