Diophantine approximation with integers having no large prime factors

Kunjakanan Nath, Habibur Rahaman

Given any irrational number $\alpha$, we show that for any $0<\theta <6/17$, there are infinitely many $y$-smooth (friable) numbers $n$ such that $$\Vert n\alpha \Vert <n^{-\theta },$$ where $(\log n)^C\leq y\…