This paper introduces some methods to determine the simultaneous approximation constants of a class of well approximable numbers ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$. The approach relies on results on the connection between the set of all $s$-adic expansions ($s\ge 2$) of ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$ and their associated approximation constants. As an application, explicit construction of real numbers ${\zeta }_1,{\zeta }_2,...,{\zeta }_k$ with prescribed approximation properties are deduced and illustrated by Matlab plots.

In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.

Introduction. Soit θ un élément de ¹=ℝ/ℤ. Considérons la suite des multiples de θ, $x={\left(n\theta \right)}_{n\in \mathbb{N}}$. Pour tout n ∈ ℕ, ordonnons les n+1 premiers termes de cette suite, 0 = y₀ ≤ y₁ ≤...≤ yₙ ≤ 1 = pθ, p=0,...,n. La suite (y₀,...,yₙ) découpe l’intervalle [0,1] en n+1 intervalles qui ont au plus trois longueurs distinctes, la plus grande de ces longueurs étant la somme des deux autres. Cette propriété a été conjecturé par Steinhaus, elle est étroitement liée au développement en fraction continue de θ. On peut aussi...