On the Distribution of αp2 + β Modulo One and r-Free Integers

Let α be an irrational number, β a real number, and a1,…,as a set of distinct positive integers that do not form a reduced residue system modulo pr for any prime p. In this work, we establish that there are infinitely many prime numbers p that satisfy the condition ||αp2+β||<p−θ for a suitable θ, and at the same time each of the numbers p+a1,…, p+as is r-free.