We obtain a generalization of Furstenberg’s Diophantine Theorem on non-lacunary multiplicative semigroups. For example we show that the sets of sums
{
(
p
1
n
q
1
m
+
p
2
n
q
2
m
)
α
:
n
,
m
∈
N
}
\{(p_1^nq_1^m + p_2^nq_2^m)\alpha :n,m \in \mathbb {N}\}
and
{
(
p
1
n
q
1
m
+
2
n
)
α
:
n
,
m
∈
N
}
\{(p_1^nq_1^m + 2^n)\alpha : n,m \in \mathbb {N}\}
are dense in the circle
T
=
R
/
Z
\mathbb {T} = \mathbb {R}/ \mathbb {Z}
for all irrational
α
\alpha
, where
(
p
i
,
q
i
)
(p_i, q_i)
are distinct pairs of multiplicatively independent integers for
i
=
1
,
2
i=1, 2
.