A lower bound for the approximation of \(\ln 2\) by rational numbers. (Russian) Zbl 0635.10025

Rukhadze, E. A.

Vestn. Mosk. Univ., Ser. I 1987, No. 6, 25-29 (1987).

Let \(u\in [0,1]\), and let \(\varepsilon\) be any positive number. It is proved that for any positive integers \(p,q>q_0(\varepsilon,u)\) we have
\[ | q \ln 2-p| > q^{-\gamma -\varepsilon}, \] where \(\gamma =\gamma(u)\) is a complicated function of \(u\). Presumably, the best choice is \(u=1/7\) which leads to \(\gamma =2.893\).

Reviewer: Veikko Ennola (Turku)

MSC:

11J04

Homogeneous approximation to one number

Keywords:

diophantine approximation of ln 2