The natural extension of the Gauss map and the Hermite best approximations

Following Humbert and Lagarias, given a real number θ, we call a nonzero vector (p, q) ∈ Z × N a Hermite best approximation vector of θ if it minimizes a quadratic form f∆(x, y) = (x− yθ)2 + y 2 ∆ for at least one real number ∆ > 0. Hermite observed that if (p, q) is such a minimum with q > 0, then the fraction p/q must be a convergent of the continued fraction expansion of θ. Using minimal vectors in lattices, we give new proofs of some results of Humbert and Meignen and complete their works. In particular, we show that the proportion of Hermite best approximation vectors among convergents is almost surely ln 3/ ln 4. The main tool of the proofs is the natural extension of the Gauss map x ∈ ]0, 1[→ {1/x}.