Diophantine approximation with integers having no large prime factors

TL;DR

This paper proves that for any irrational number, infinitely many smooth integers approximate it within a certain error, improving previous bounds using advanced exponential sum techniques.

Contribution

It advances Diophantine approximation by establishing new bounds for smooth numbers approximating irrationals, surpassing prior results.

Findings

Infinitely many smooth numbers satisfy the approximation inequality for θ<6/17.

Improves previous exponent bounds from 1/3 to 6/17 for smooth number approximations.

Utilizes dispersion method and bounds on Kloosterman sums over smooth numbers.

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.