Abstract
Let
θ be a real number satisfying
1
<
θ
<
2
, and let
A
(
θ
)
be the set of polynomials with coefficients in
{
0
,
1
}
, evaluated at
θ. Using a result of Bugeaud, we prove by elementary methods that
θ is a Pisot number when the set
(
A
(
θ
)
−
A
(
θ
)
−
A
(
θ
)
)
is discrete; the problem whether Pisot numbers are the only numbers
θ such that 0 is not a limit point of
(
A
(
θ
)
−
A
(
θ
)
)
is still unsolved. We also determine the three greatest limit points of the quantities
inf
{
c
,
c
>
0
,
c
∈
C
(
θ
)
}
, where
C
(
θ
)
is the set of polynomials with coefficients in
{
−
1
,
1
}
, evaluated at
θ, and we find in particular infinitely many Perron numbers
θ such that the sets
C
(
θ
)
are discrete.