On the distribution of $\alpha p^2$ modulo one with prime $p$ of a special form

Abstract

Let ${\mathcal{P}}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for $\alpha \in \mathbb{R}\backslash \mathbb{Q},\beta \in \mathbb{R}$ and $0<\theta <10/1561$, there exist infinitely many primes $p$, such that $\Vert \alpha p^2+\beta \Vert <p^{-\theta }$ and $p+2={\mathcal{P}}_4$, which constitutes an improvement upon the previous result.

30 pages