Topological properties of two-dimensional number systems

Akiyama, Shigeki; Thuswaldner, Jörg M.

Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 1, pp. 69-79.

Pour une matrice réelle $M$ d’ordre $2$ donnée, on peut définir la notion de représentation $M$-adique d’un élément de ${ℝ}^2$. On note $ℱ$ le domaine fondamental constitué des nombres de ${ℝ}^2$ dont le développement “$M$-adique” ne commence pas par $0$. C’est l’analogue dans ${ℝ}^2$ des nombres $q$-adiques, où la matrice $M$ joue le rôle de la base $q$. Kátai et Környei ont démontré que $ℱ$ est compact, et que ${ℝ}^2$ s’écrit comme la réunion dénombrable de certains translatés de $ℱ$, l’intersection de $2$ quelconques d’entre eux étant de mesure nulle. Dans cet article, nous construisons des points qui appartiennent simultanément à trois translatés de $ℱ$, et nous montrons que $ℱ$ est connexe. Nous donnons aussi une propriété sur la structure des points intérieurs de $ℱ$.

In the two dimensional real vector space ${ℝ}^2$ one can define analogs of the well-known $q$-adic number systems. In these number systems a matrix $M$ plays the role of the base number $q$. In the present paper we study the so-called fundamental domain $ℱ$ of such number systems. This is the set of all elements of ${ℝ}^2$ having zero integer part in their “$M$-adic” representation. It was proved by Kátai and Környei, that $ℱ$ is a compact set and certain translates of it form a tiling of the ${ℝ}^2$. We construct points, where three different tiles of this tiling coincide. Furthermore, we prove the connectedness of $ℱ$ and give a result on the structure of its inner points.

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