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 $\theta < 1/1000$.
Citation
San-Ying Shi. "On the distribution of $\alpha p$ modulo one for primes $p$ of a special form." Osaka J. Math. 49 (4) 993 - 1004, December 2012.
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