  We prove the density of the sets of the form ${{\lambda}_1^m {\mu}_1^n
{\xi}_1 +...+{\lambda}_k^m {\mu}_k^n {\xi}_k : m,n \in \mathbb N}$ modulo one,
where $\lambda_i$ and $\mu_i$ are multiplicatively independent algebraic
numbers satisfying some additional assumptions. The proof is based on analysing
dynamics of higher-rank actions on compact abelean groups.
