Algebraic numbers and density modulo 1

Let lambda(1), mu(1) and lambda(2), mu(2) 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 lambda(i), mu(i) and their conjugates, we prove that for any real numbers xi(1), xi(2), with at least one xi(i) not equal 0, the set {lambda(n)(1)mu(m)(1)xi(1) + lambda(n)(2)mu(m)(2)xi(2): n,m is an element of N} is dense modulo 1. (c) 2007 Elsevier Inc. All rights reserved.