On the distribution of $\alpha p$ modulo one for primes $p$ of a special form

Abstract

In this paper it is proved that for any irrational $\alpha$ and some $0 < \theta \le 1.5/100$, there are infinitely many primes $p$ such that $p+2$ has at most two prime factors and $\lVert\alpha p+\beta\rVert < p^{-\theta}$ which improves K. Matomäki's result $\theta < 1/1000$.