 
Summary: TOPOLOGICAL PROPERTIES OF TWODIMENSIONAL NUMBER
SYSTEMS
SHIGEKI AKIYAMA AND J
ORG M. THUSWALDNER
Abstract. In the two dimensional real vector space R 2 one can dene analogs of the
wellknown qadic number systems. In these number systems a matrix M plays the role of
the base number q. In the present paper we study the socalled fundamental domain F of
such number systems. This is the set of all elements of R 2 having zero integer part in their
\Madic" representation. It was proved by Katai and K}ornyei, that F is a compact set and
certain translates of it form a tiling of the R 2 . We construct points, where three dierent
tiles of this tiling coincide. Furthermore, we prove the connectedness of F and give a result
on the structure of its inner points.
1. Introduction
In this paper we use the following notations: R, Q , Z and N denote the set of real numbers,
rational numbers, integers and positive integers, respectively. If x 2 R we will write bxc
for the largest integer less than or equal to x. will denote the 2dimensional Lebesgue
measure. Furthermore, we write @A for the boundary of the set A and int(A) for its interior.
diag( 1 ; 2 ) denotes a 2 2 diagonal matrix with diagonal elements 1 and 2 .
Let q 2 be an integer. Then each positive integer n has a unique qadic representation
of the shape n =
