On a theorem of Borel on diophantine approximation

Abstract

A theorem of É. Borel’s asserts that one of any three consecutive convergents of a real number a, which we denote \(\frac{p}{q}\), satisfies the inequality \(\left| a-\frac{p}{q} \right| < \frac{C}{q^2}\) with \(C=\frac{1}{\sqrt{5}}\). In this paper we give more precise information about the constant C.