Algebraic Numbers and Density Modulo 1, Ii

This is a companion paper to [8]. In [8], using ideas of Berend [3] and Kra [6], it was proved that the sets of the form {λn 1 µm 1 ξ1 + λn 2 µm 2 ξ2 : n, m ≥ 1}, where ξ1, ξ2 ∈ R, λ1, µ1 and λ2, µ2 are two pairs of multiplicatively independent real algebraic numbers satisfying certain technical conditions, including that µi ∈ Q(λi), i = 1, 2, are dense modulo 1/κ, for some κ ≥ 1. In this paper we extend the result from [8], showing that the condition µi ∈ Q(λi) can be removed by imposing appropriate conditions on the norms of con- jugates of λi, µi and the degree of the algebraic numbers λn i µm i .