Diophantine approximation with primes in an arithmetic progression

Let $α\in \mathbb{R} \setminus \mathbb{Q}$, $β\in \R$, $N \in \mathbb{R}_{\ge 1}$ and $ Δ\in (0, 1/2)$. For any real $y$, let $\|y\|$ denote the distance from $y$ to the nearest integer. In the first part of this paper, we show that given two coprime integers $u, v \ge 1$, there are infinitely man