Algebraic numbers, hyperbolicity, and density modulo one

We prove the density of the sets of the form{λ1mμ1nξ1+⋯+λkmμknξk:m,n∈N} modulo one, where λiλi and μiμi are multiplicatively independent algebraic numbers satisfying some additional assumptions. The proof is based on analysing dynamics of higher-rank actions on compact abelian groups.