Abstract
Properties of Pisot numbers have long been of interest. One line of questioning, initiated by Erdös, Joò and Komornik in (Bull Soc Math France 118:377–390,
1990
), is the study of the set
Λ
m
(
β
)
the spectrum of
β
and the determination of
l
m
(
β
)
for Pisot number
β
, where
Λ
m
(
β
)
denotes the set of numbers having at least one representation of the form
ω
=
ε
n
β
n
+
ε
n
-
1
β
n
-
1
+
⋯
+
ε
1
β
+
ε
0
,
such that the
ε
i
∈
{
-
m
,
…
,
0
,
…
,
m
}
, for all
0
≤
i
≤
n
, and
l
m
(
β
)
=
inf
{
|
ω
|
:
ω
∈
Λ
m
,
ω
≠
0
}
.
In this paper, we consider
Λ
m
(
β
)
, where
β
is a formal power series over a finite field and the
ε
i
are polynomials of degree at most
m
for all
0
≤
i
≤
n
. Our main result is to give a full answer in the Laurent series case, to an old question of Erdős and Komornik (Acta Math Hungar 79:57–83,
1998
), as to whether
l
1
(
β
)
=
0
for all non-Pisot numbers. More generally, we characterize the inequalities
l
m
(
β
)
>
0
.