On the distribution of $\alpha p^2$ 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 $\|x\|$ 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{equation*} \|\alpha p^2+\beta\|<p^{-\frac{14(\gamma_1+\gamma_2)-27}{43}+\varepsilon}, \end{equation*} provided that $27/14<\gamma_1+\gamma_2<2$. This result constitutes an generalization upon the previous result of Dimitrov [4].