On a Theorem of Legendre on Diophantine Approximation

Jaroslav Hančl, Tho Phuoc Nguyen

Published 2024-07-15Version 1

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.