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Best approximation of an element of the torus \(\mathbb{T}^ 2\)and geometry of the sequence of multiples of this element. (Meilleures approximations d’un élément du tore \(\mathbb{T}^ 2\) et géométrie de la suite des multiples de cet élément.) (French) Zbl 0863.11043
Let \(\mathbb{T}^d\) denote the \(d\)-dimensional torus. Define the sequence \((q_n)\) of best approximations of \(\theta\in \mathbb{T}^d\) by \(q_1= 1\) and \(q_{n+1}= \min\{ k\in \mathbb{N}\), \(k> q_n\) and \(|k\theta |< |q_n\theta |\}\). We prove two results about the sequence \((q_n)\):
(1) Let \(\theta\in \mathbb{T}^2\). If \(\mathbb{N}\theta\) is dense in \(\mathbb{T}^2\) then \(q_{n+1} |q_n \theta |\;|q_{n-1} \theta|\geq {1\over {100}}\) for infinitely many \(n\).
(2) The sequence \(({{|q_{n-1} \theta|} \over {|q_n \theta|}})\) is unbounded for almost all \(\theta\in \mathbb{T}^2\).
Using Voronoï diagrams we have partially generalized to \(\mathbb{T}^2\) the following property: For all \(\theta\in \mathbb{T}^1\) and all \(n\in \mathbb{N}\), \(\mathbb{T}^1 \setminus \{0, \theta, \dots, n\theta\}\) is composed of \((n+1)\) intervals whose lengths have at most three distinct values.

MSC:
11J13 Simultaneous homogeneous approximation, linear forms
11J70 Continued fractions and generalizations
11K60 Diophantine approximation in probabilistic number theory
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