On a Theorem of Legendre on Diophantine Approximation

Jaroslav Hančl; Tho Phuoc Nguyen

On a Theorem of Legendre on Diophantine Approximation

Number Theory 2024-07-17 v1

Authors: Jaroslav Hančl , Tho Phuoc Nguyen

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.

Keywords

Diophantine approximation Legendre's theorem Chernoff bound Cramér lower bound Convergent

Cite

@article{arxiv.2407.11166,
  title  = {On a Theorem of Legendre on Diophantine Approximation},
  author = {Jaroslav Hančl and Tho Phuoc Nguyen},
  journal= {arXiv preprint arXiv:2407.11166},
  year   = {2024}
}

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