Multiplicatively independent integers and dense modulo $1$ sets of sums

Let $c \in \mathbb{R},$ $c > 0,$ $\beta \in \mathbb{R}$ and $a_1 > a_2 > 1$ and $b_1 > b_2 > 1$ be two distinct pairs of multiplicatively independent integers.If $b_1 > a_1$ and $a_2 > b_2$ or $b_1 < a_1$ and $a_2 < b_2$ then we prove that for every $\xi_1, \xi_2,$ with at least one $\xi_i$ irrational, there exists $q \in \mathbb{N}$ such that the set of sums {a_1^ma_2^{n}q \xi_1+b_1^mb_2^{n}q \xi_2+c^{m+n} \beta:m,n \in \mathbb{N}}, is dense modulo $1$ for all reals $\beta.$