Diophantine approximation with integers having no large prime factors

Given any irrational number $\alpha$, we show that for any $0<\theta <6\mathrm{/}17$, there are infinitely many $y$-smooth (friable) numbers $n$ such that $$\mathrm{\parallel }n\alpha \mathrm{\parallel }<n^{-\theta },$$ where $(\log n{)}^C\le y\le n$ for some large constant $C>0$. This improves the previous work of Baker, who obtained the exponent $1\mathrm{/}3-2\mathrm{/}(3C)+o(1)$ in the case of $y\ge (\log n{)}^C$, and that of Yau, who obtained the exponent $1\mathrm{/}3$ when $y=n^{o(1)}$. Our proof is based on the dispersion method together with arithmetic inputs coming from the average bounds for Kloosterman sums over smooth numbers.