Let
S
d
(
α
)
{S_d}(\alpha )
denote the set of all integers which can be expressed in the form
∑
ε
i
[
α
i
]
\sum {{\varepsilon _i}[{\alpha ^i}]}
, with
ε
i
∈
{
0
,
…
,
d
−
1
}
{\varepsilon _i} \in \{ 0, \ldots ,d - 1\}
, where
d
≥
2
d \geq 2
is an integer and
α
≥
1
\alpha \geq 1
is real, and let
I
d
{I_d}
denote the set of
α
\alpha
so that
S
d
(
α
)
=
Z
+
{S_d}(\alpha ) = {{\mathbf {Z}}^ + }
. We show that
I
d
=
[
1
,
r
d
)
∪
{
d
}
{I_d} = [1,{r_d}) \cup \{ d\}
, where
r
2
=
13
1
/
4
,
r
3
=
22
1
/
3
{r_2} = {13^{1/4}},{r_3} = {22^{1/3}}
and
r
2
=
(
d
2
−
d
−
2
)
1
/
2
{r_2} = {({d^2} - d - 2)^{1/2}}
for
d
≥
4
d \geq 4
. If
α
∉
I
d
\alpha \notin {I_d}
we show that
T
d
(
α
)
{T_d}(\alpha )
, the complement of
S
d
(
α
)
{S_d}(\alpha )
, is infinite, and discuss the density of
T
d
(
α
)
{T_d}(\alpha )
when
α
>
d
\alpha > d
. For
d
≥
4
d \geq 4
and a particular quadratic irrational
β
=
β
(
d
)
>
d
\beta = \beta (d) > d
, we describe
T
d
(
β
)
{T_d}(\beta )
explicitly and show that
|
T
d
(
β
)
∩
[
0
,
n
]
|
|{T_d}(\beta ) \cap [0,n]|
is of order
n
e
(
d
)
{n^{e(d)}}
, where
e
(
d
)
>
1
e(d) > 1
.