On a Theorem of Legendre on Diophantine Approximation

TL;DR

This paper refines Legendre's theorem on Diophantine approximation, improving bounds on irreducible fractions close to a real number, building upon previous enhancements by Barbolosi and Jager.

Contribution

It provides a sharper version of Legendre's theorem, extending the bounds for when fractions approximate real numbers, advancing the understanding of Diophantine approximation.

Findings

Refined bounds for irreducible fractions approximating real numbers

Improved theorems over previous results by Barbolosi and Jager

Enhanced criteria for convergents in Diophantine approximation

Abstract

Legendre's theorem states that every irreducible fraction $\frac{p}{q}$ which satisfies the inequality $\left |\alpha-\frac{p}{q} \right | < \frac{1}{2q^2}$ is convergent to $\alpha$. Later Barbolosi and Jager improved this theorem. In this paper we refine these results.