On a Theorem of Nathanson on Diophantine Approximation

TL;DR

This paper refines Nathanson's 1974 theorem on Diophantine approximation, providing improved bounds for approximating irrationals with continued fractions having large elements.

Contribution

The paper improves Nathanson's original bounds on Diophantine approximation for irrationals with continued fractions.

Findings

Refined approximation bounds for irrationals with large continued fraction elements

Enhanced understanding of the relationship between continued fraction elements and approximation quality

Potential applications in number theory and Diophantine approximation

Abstract

In 1974, M. B. Nathanson proved that every irrational number $\alpha$ represented by a simple continued fraction with infinitely many elements greater than or equal to $k$ is approximable by an infinite number of rational numbers $p/q$ satisfying $|\alpha-p/q|<1/(\sqrt{k^2+4}q^2)$. In this paper we refine this result.