Topological properties of two-dimensional number systems

Shigeki Akiyama; Jörg M. Thuswaldner

Shigeki Akiyama ; Jörg M. Thuswaldner

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

Résumé
Abstract

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.

MR Zbl

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     author = {Shigeki Akiyama and J\"org M. Thuswaldner},
     title = {Topological properties of two-dimensional number systems},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {69--79},
     publisher = {Universit\'e Bordeaux I},
     volume = {12},
     number = {1},
     year = {2000},
     zbl = {1012.11072},
     mrnumber = {1827838},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2000__12_1_69_0/}
}

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Shigeki Akiyama; Jörg M. Thuswaldner. Topological properties of two-dimensional number systems. Journal de théorie des nombres de Bordeaux, Tome 12 (2000) no. 1, pp. 69-79. https://jtnb.centre-mersenne.org/item/JTNB_2000__12_1_69_0/

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