Algebraic numbers and density modulo 1

Let λ1,μ1 and λ2,μ2 be two pairs of rationally independent real algebraic numbers of degree 2, with absolute values greater than 1, such that the absolute values of their conjugates are also greater than 1. Under some additional assumptions, on the relation between λi,μi and their conjugates, we prove that for any real numbers ξ1,ξ2, with at least one ξi≠0, the set is dense modulo 1.