 
Summary: HEIGHT REDUCING PROBLEM ON ALGEBRAIC
INTEGERS
SHIGEKI AKIYAMA, PAULIUS DRUNGILAS, AND JONAS JANKAUSKAS
Abstract. Let be an algebraic integer and assume that it is expanding,
i.e., its all conjugates lie outside the unit circle. We show several results
of the form Z[] = B[] with a certain finite set B Z. This property is
called height reducing property, which attracted special interest in the self
affine tilings. Especially we show that if is quadratic or cubic trinomial,
then one can choose B = {0, ±1, . . . , ± (N()  1)}, where N() stands
for the absolute norm of over Q.
1. Introduction
Let be an algebraic integer with conjugates 1 = , 2, . . . , d lying
outside the unit circle (including itself). Such numbers are called expanding
algebraic numbers. We are interested in the height reducing property of ,
that is
Z[] = B[]
for a certain finite set B Z. We note that
Lemma 1. If an algebraic integer ,  > 1, has height reducing property,
then is expanding.
Proof. Suppose has height reducing property with a finite set B Z. First
