 
Summary: Self aĆne tiling and Pisot numeration system
Shigeki Akiyama
1 Introduction
Let > 1 be a real number. Consider an expansion of positive real number x:
x = aN N + aN 1 N 1 + aN 2 N 2 + ;
with a i 2 Z \ [0; ). A greedy expansion of x in base is such expansion with
jx
N
X
M
a i i j < M (1)
for any M . By using greedy algorithm, such an expansion always exists for any x. This is
a natural extension of decimal or binary expansion to a real base. An expansion of x is
admissible, if (1) holds for all M . Hereafter, we use a notation
x = aN aN 1 aN 2 :
A Pisot number is an algebraic integer greater than 1 whose conjugates other than
itself have modulus smaller than 1. We have a particular interest in the case when > 1
is a Pisot number. Surprisingly, one can nd many similar phenomena with binary or
decimal expansion. See [1]. We use a term 'Pisot numeration system' to call this method
to represent real numbers in a power series in Pisot number base.
