 
Summary: Discrete Mathematics and Theoretical Computer Science DMTCS vol. 7, 2005, 269312
Connectedness of number theoretic tilings
Shigeki Akiyama1
and Nertila Gjini2
1
Department of Mathematics, Faculty of Science, Niigata University, Ikarashi 28050, Niigata 9502181, Japan,
email address: akiyama@math.sc.niigatau.ac.jp
2
Department of Mathematics, University of New York Tirana, Rr. Komuna e Parisit, Tirana, Albania,
email address: ngjini@unyt.edu.al
received May 26, 2003, accepted Nov 14, 2005.
Let T = T(A, D) be a selfaffine tile in Rn
defined by an integral expanding matrix A and a digit set D. In connection
with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers
{0, 1, . . . ,  det(A)  1}. It is shown that in R3
and R4
, for any integral expanding matrix A, T(A, D) is connected.
We also study the connectedness of Pisot dual tilings which play an important role in the study of expansion,
substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is
arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However
