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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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