Topological properties of two-dimensional number systems

Akiyama, Shigeki, and Thuswaldner, Jörg M.. "Topological properties of two-dimensional number systems." Journal de théorie des nombres de Bordeaux 12.1 (2000): 69-79. <http://eudml.org/doc/248483>.

@article{Akiyama2000,
abstract = {In the two dimensional real vector space $\mathbb \{R\}^2$ 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 $\mathcal \{F\}$ of such number systems. This is the set of all elements of $\mathbb \{R\}^2$ having zero integer part in their “$M$-adic” representation. It was proved by Kátai and Környei, that $\mathcal \{F\}$ is a compact set and certain translates of it form a tiling of the $\mathbb \{R\}^2$. We construct points, where three different tiles of this tiling coincide. Furthermore, we prove the connectedness of $\mathcal \{F\}$ and give a result on the structure of its inner points.},
author = {Akiyama, Shigeki, Thuswaldner, Jörg M.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {radix representation; fundamental domain; arcwise connected},
language = {eng},
number = {1},
pages = {69-79},
publisher = {Université Bordeaux I},
title = {Topological properties of two-dimensional number systems},
url = {http://eudml.org/doc/248483},
volume = {12},
year = {2000},
}

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
PB - Université Bordeaux I
VL - 12
IS - 1
SP - 69
EP - 79
AB - In the two dimensional real vector space $\mathbb {R}^2$ 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 $\mathcal {F}$ of such number systems. This is the set of all elements of $\mathbb {R}^2$ having zero integer part in their “$M$-adic” representation. It was proved by Kátai and Környei, that $\mathcal {F}$ is a compact set and certain translates of it form a tiling of the $\mathbb {R}^2$. We construct points, where three different tiles of this tiling coincide. Furthermore, we prove the connectedness of $\mathcal {F}$ and give a result on the structure of its inner points.
LA - eng
KW - radix representation; fundamental domain; arcwise connected
UR - http://eudml.org/doc/248483
ER -