 
Summary: ON THE BOUNDARY OF SELF AFFINE TILINGS GENERATED
BY PISOT NUMBERS
SHIGEKI AKIYAMA
Abstract. Denition and fundamentals of tilings generated by Pisot numbers
are shown by arithmetic consideration. Results include the case that a Pisot
number does not have a nitely expansible property, i.e. a soc Pisot case.
Especially we show that the boundary of each tile has Lebesgue measure zero
under some weak condition.
1. Introduction
First we explain notations used in this paper. The rational integers is denoted by
Z, the rational numbers by Q, the complex numbers by C and the positive integers
by N. We denote by Z[u], the ring generated by Z and u 2 C , and by Q(u), the
minimum eld containing Q and u. We write A for the subset of A with constraints
by its subscript ` ', when the subscript is a conditional term. For example, Z` is
the integers not less than `. Let > 1 be a real number which is not an integer. A
greedy expansion of a positive real x in base is an expansion of a form:
x =
1
X
i=N0
