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 $\Vert x\Vert$ 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{equation*} \|αp^2+β\|<p^{-\frac{14(γ_1+γ_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].

13 pages