On the distribution of $\alpha p^{\gamma }+\beta$ modulo one

Let $\mathrm{\parallel }\cdot \mathrm{\parallel }$ denote the minimum distance to an integer. For $0<\gamma <1$, $\theta >0$ and $(\alpha, \beta) \in $\mathbb{R}$ \setminus {0} \times $\mathbb{R}$$ we study when \begin{equation*} |\alpha p^{\gamma}+\beta |<p^{-\theta}, \end{equation*} holds for infinitely many primes $p$ of a special type. In particular, we consider when this inequality holds for primes $p$ such that $p+2$ has few prime factors counted with multiplicity. This is done using an exponential sum estimate of the author and the linear sieve of Iwaniec with bilinear error term. This is related to recent work of Tolev, Todorova, Mat"{o}maki and Cai.