On a Diophantine Inequality with Primes Yielding Square-Free Sums with Given Numbers

Let $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ and $\beta \in \mathbb{R}$ be given. Suppose that $a_1,\dots ,a_s$ are distinct positive integers that do not contain a reduced residue system modulo $p^2$ for any prime $p$. We prove that there exist infinitely many primes $p$ such that the inequality $\mathrm{\mid }\mathrm{\mid }\alpha p+\beta \mathrm{\mid }\mathrm{\mid }<p^{-1\mathrm{/}10}$ holds and all the numbers $p+a_1,\dots ,p+a_s$ are square-free.