Summary: TOPOLOGICAL PROPERTIES OF TWO-DIMENSIONAL NUMBER
SHIGEKI AKIYAMA AND J¨ORG M. THUSWALDNER
Abstract. In the two dimensional real vector space R2
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 F of
such number systems. This is the set of all elements of R2
having zero integer part in their
"M-adic" representation. It was proved by K´atai and Kornyei, that F is a compact set and
certain translates of it form a tiling of the R2
. We construct points, where three different
tiles of this tiling coincide. Furthermore, we prove the connectedness of F and give a result
on the structure of its inner points.
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 R we will write x
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.