Diophantine approximation with integers having no large prime factors

Abstract: Given any irrational number $α$ and a real number $κ>0$, we show that for any $θ<θ_0(y)$, there are infinitely many $y$-smooth (friable) numbers $n$ such that $$|nα|<n^{-θ},$$ where $θ_0(y)=6/17$ if $\exp((\log\log n)^{2+κ})\leq y\leq n^{o(1)}$ and $θ_0(y)=20/59$ 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.