A generalization of Furstenberg's Diophantine Theorem

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