Pour une matrice réelle d’ordre donnée, on peut définir la notion de représentation -adique d’un élément de . On note le domaine fondamental constitué des nombres de dont le développement “-adique” ne commence pas par . C’est l’analogue dans des nombres -adiques, où la matrice joue le rôle de la base . Kátai et Környei ont démontré que est compact, et que s’écrit comme la réunion dénombrable de certains translatés de , l’intersection de 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 .
In the two dimensional real vector space one can define analogs of the well-known -adic number systems. In these number systems a matrix plays the role of the base number . In the present paper we study the so-called fundamental domain of such number systems. This is the set of all elements of having zero integer part in their “-adic” representation. It was proved by Kátai and Környei, that is a compact set and certain translates of it form a tiling of the . 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.
@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}, pages = {69--79}, publisher = {Universit\'e Bordeaux I}, volume = {12}, number = {1}, year = {2000}, mrnumber = {1827838}, zbl = {1012.11072}, language = {en}, url = {http://archive.numdam.org/item/JTNB_2000__12_1_69_0/} }
TY - JOUR AU - Akiyama, Shigeki AU - Thuswaldner, Jörg M. TI - Topological properties of two-dimensional number systems JO - Journal de théorie des nombres de Bordeaux PY - 2000 SP - 69 EP - 79 VL - 12 IS - 1 PB - Université Bordeaux I UR - http://archive.numdam.org/item/JTNB_2000__12_1_69_0/ LA - en ID - JTNB_2000__12_1_69_0 ER -
%0 Journal Article %A Akiyama, Shigeki %A Thuswaldner, Jörg M. %T Topological properties of two-dimensional number systems %J Journal de théorie des nombres de Bordeaux %D 2000 %P 69-79 %V 12 %N 1 %I Université Bordeaux I %U http://archive.numdam.org/item/JTNB_2000__12_1_69_0/ %G en %F 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, Tome 12 (2000) no. 1, pp. 69-79. http://archive.numdam.org/item/JTNB_2000__12_1_69_0/
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