Diophantine approximation with integers having no large prime factors

Given any irrational number $\alpha$ and a real number $\kappa>0$, we show that for any $\theta<\theta_0(y)$, there are infinitely many $y$-smooth (friable) numbers $n$ such that

$$\|n\alpha\|<n^{-\theta},$$

where $\theta_0(y)=6/17$ if $\exp((\log\log n)^{2+\kappa})\leq y\leq n^{o(1)}$ and $\theta_0(y)=20/59$ if $n^{o(1)}<y\leq n$. Here $\|x\|$ denotes the distance from $x$ to the nearest integer. Our proof is based on the dispersion method together with arithmetic inputs coming from the average bounds for Kloosterman sums over smooth numbers.