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

Abstract: Let $\alpha\in \mathbb{R}\setminus\mathbb{Q}$ and $\beta\in \mathbb{R}$ be given. Suppose that $a_1,\ldots,a_s$ 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 $||\alpha p+\beta||<p^{-1/10}$ holds and all the numbers $p+a_1,\ldots,p+a_s$ are square-free.