Distribution of α p 2 Modulo One with Prime Variable 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 $\alpha \in \left(ℝ/\mathbb{Q}\right),\beta \in ℝ$ , and $0<\theta <\left(10/1561\right)$ , there exist infinitely many primes $p$ , such that $‖\alpha p^2+\beta ‖<p^{-\theta }$ and $p+2={\mathcal{P}}_4$ , which constitutes an improvement upon the previous result.