# Topological properties of two-dimensional number systems

Shigeki Akiyama; Jörg M. Thuswaldner

Journal de théorie des nombres de Bordeaux (2000)

- Volume: 12, Issue: 1, page 69-79
- ISSN: 1246-7405

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topAkiyama, 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 -

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