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Summary: TOPOLOGICAL PROPERTIES OF TWO-DIMENSIONAL NUMBER
SYSTEMS
SHIGEKI AKIYAMA AND J
ORG M. THUSWALDNER
Abstract. In the two dimensional real vector space R 2 one can de ne 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 F of
such number systems. This is the set of all elements of R 2 having zero integer part in their
\M-adic" 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 di erent
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 2-dimensional 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 q-adic representation
of the shape n =
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