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

Let \(\lfloor t\rfloor \) $\lfloor t\rfloor$ denote the integer part of \(t\in \mathbb {R}\) $t\in \mathbb{R}$ and \(\Vert x\Vert \) $\Vert x\Vert$ the distance from x to the nearest integer. Suppose that \(1/2<\gamma _2<\gamma _1<1\) $1/2<{\gamma }_2<{\gamma }_1<1$ are two fixed constants. In this paper, it is proved that, whenever \(\alpha \) $\alpha$ is an irrational number and \(\beta \) $\beta$ is any real number, there exist infinitely many prime numbers p in the intersection of two Piatetski–Shapiro sets, i.e., \(p=\lfloor n_1^{1/\gamma _1}\rfloor =\lfloor n_2^{1/\gamma _2}\rfloor \) $p=\lfloor n_1^{1/{\gamma }_1}\rfloor =\lfloor n_2^{1/{\gamma }_2}\rfloor$ , such that \(\begin{aligned} \Vert \alpha p+\beta \Vert <p^{-\frac{12(\gamma _1+\gamma _2)-23}{38}+\varepsilon }, \end{aligned}\) $$\begin{array}{c}\hfill \Vert \alpha p+\beta \Vert <p^{-\frac{12({\gamma }_1+{\gamma }_2)-23}{38}+\epsilon },\end{array}$$ provided that \(23/12<\gamma _1+\gamma _2<2\) $23/12<{\gamma }_1+{\gamma }_2<2$ . This result constitutes an generalization upon the previous result of Dimitrov (Indian J Pure Appl Math 54(3):858–867, 2023).