The natural extension of the Gauss map and Hermite best approximations

TL;DR

This paper studies Hermite best approximation vectors of real numbers, showing their proportion among convergents is ln 3/ ln 4, using the natural extension of the Gauss map to analyze their distribution.

Contribution

It introduces the natural extension of the Gauss map to determine the asymptotic proportion of Hermite best approximation vectors among convergents.

Findings

Proportion of Hermite best approximation vectors among convergents is ln 3/ ln 4.

Uses the natural extension of the Gauss map for analysis.

Provides a measure-theoretic result on approximation vectors.

Abstract

Hermite best approximation vectors of a real number $\theta$ were introduced by Lagarias. A nonzero vector (p, q) $\in$ Z x N is a Hermite best approximation vector of $\theta$ if there exists $\Delta$ > 0 such that (p -- q$\theta$) 2 + q 2 /$\Delta$ $\le$ (a -- b$\theta$) 2 + b 2 /$\Delta$ for all nonzero (a, b) $\in$ Z 2. Hermite observed that if q > 0 then the fraction p/q must be a convergent of the continued fraction expansion of $\theta$ and Lagarias pointed out that some convergents are not associated with a Hermite best approximation vectors. In this note we show that the almost sure proportion of Hermite best approximation vectors among convergents is ln 3/ ln 4. The main tool of the proof is the natural extension of the Gauss map x $\in$]0, 1[$\rightarrow$ {1/x}.