Distribution of \(\alpha p^2\) modulo one with prime variable \(p\) of a special form (Q2034572)

Summary: 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 \(\alpha\in (\mathbb{R}/\mathbb{Q})\), \(\beta\in\mathbb{R}\), and \(0<\theta<(10/1561)\), 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.