Diophantine approximation with smooth numbers

TL;DR

This paper establishes new bounds on approximating irrational numbers using rational numbers with smooth denominators, improving previous results by employing exponential sums tailored to smooth numbers.

Contribution

It introduces a novel exponential sum technique over smooth numbers, enhancing approximation bounds and approaching an exponent of 1/3.

Findings

New approximation bounds approaching an exponent of 1/3

Use of exponential sums over smooth numbers improves results

Strengthened previous theorems with refined techniques

Abstract

We prove a theorem about approximation to an irrational number by rational numbers whose denominator n is free of prime factors bigger than a power of log n. We strengthen the result in version 1 by using an exponential sum over smooth numbers tailored to the application. The new exponent approaches 1/3 as the exponent of log n becomes large.