Diophantine approximation with integers having no large prime factors

Abstract

Given any irrational number $\hat{I}\pm$, we show that for any $0<\hat{I}¸<6/17$, there are infinitely many $y$-smooth (friable) numbers $n$ such that $$\Vert n\hat{I}\pm \Vert <n^{-\hat{I}¸},$$ 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 $1/3-2/(3C)+o(1)$ in the case of $y\ge (\log n{)}^C$, 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.

v2. Strengthened the main result (Theorem 1) to cover the smoothness range $y\ge (\log n{)}^C$. Also, the main result holds with the exponent $6/17$ in the whole range. 32 pages; comments are welcome