Diophantine approximation with integers having no large prime factors

Abstract

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 $$\|n\alpha\| < n^{-\theta},$$ where $(\log n)^C\leq y\leq n$ for some large constant $C>0$. This improves the previous work of Baker, who obtained the exponent $1/3-2/(3C)+o(1)$ in the case of $y\geq (\log n)^C$, and that of Yau, who obtained the exponent $1/3$ 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.