Topological properties of two-dimensional number systems
Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 1, p. 69-79

In the two dimensional real vector space 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 of such number systems. This is the set of all elements of 2 having zero integer part in their “M-adic” representation. It was proved by Kátai and Környei, that is a compact set and certain translates of it form a tiling of the 2 . We construct points, where three different tiles of this tiling coincide. Furthermore, we prove the connectedness of and give a result on the structure of its inner points.

Pour une matrice réelle M d’ordre 2 donnée, on peut définir la notion de représentation M-adique d’un élément de 2 . On note le domaine fondamental constitué des nombres de 2 dont le développement “M-adique” ne commence pas par 0. C’est l’analogue dans 2 des nombres q-adiques, où la matrice M joue le rôle de la base q. Kátai et Környei ont démontré que est compact, et que 2 s’écrit comme la réunion dénombrable de certains translatés de , 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 , et nous montrons que est connexe. Nous donnons aussi une propriété sur la structure des points intérieurs de .

@article{JTNB_2000__12_1_69_0,
     author = {Akiyama, Shigeki and Thuswaldner, J\"org M.},
     title = {Topological properties of two-dimensional number systems},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {12},
     number = {1},
     year = {2000},
     pages = {69-79},
     zbl = {1012.11072},
     mrnumber = {1827838},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2000__12_1_69_0}
}
Akiyama, Shigeki; Thuswaldner, Jörg M. Topological properties of two-dimensional number systems. Journal de théorie des nombres de Bordeaux, Volume 12 (2000) no. 1, pp. 69-79. http://www.numdam.org/item/JTNB_2000__12_1_69_0/

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