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Journal Article
Numbers expressible as a difference of two Pisot numbers
2024
Overview
We characterize algebraic integers which are differences of two Pisot numbers. Each such number
α
must be real and its conjugates over
Q
must all lie in the union of the disc
|
z
|
<
2
and the strip
|
ℑ
(
z
)
|
<
1
. In particular, we prove that every real algebraic integer
α
whose conjugates over
Q
, except possibly for
α
itself, all lie in the disc
|
z
|
<
2
can always be written as a difference of two Pisot numbers. We also show that a real quadratic algebraic integer
α
with conjugate
α
′
over
Q
is always expressible as a difference of two Pisot numbers except for the cases
α
<
α
′
<
-
2
or
2
<
α
′
<
α
when
α
cannot be expressed in that form. A similar complete characterization of all algebraic integers
α
expressible as a difference of two Pisot numbers in terms of the location of their conjugates is given in the case when the degree
d
of
α
is a prime number.
Publisher
Springer International Publishing,Springer Nature B.V
Subject
