On the distribution of $$\alpha p$$ modulo one in the intersection of two Piatetski–Shapiro sets

Abstract

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).