The natural extension of the Gauss map and the Hermite best approximations (Q2086426)

Following \textit{G. Humbert} [Journ. de Math. (7) 2, 79--103 (1916; JFM 46.0271.03)] and \textit{J. C. Lagarias} [Proc. Lond. Math. Soc. (3) 69, No. 3, 464--488 (1994; Zbl 0813.11040)], the author studies the notion of so called Hermite's best approximations. Such approximations are used as a tool to complete the works by Humbert and Meignen and give new proofs of their several results. In particular, it is shown that the proportion of Hermite best approximation vectors among convergents is almost surely \(\ln 3/\ln 4\). The proofs are based on the natural extension of the classic Gauss map.