Algebraic numbers and density modulo $1,$ II

This is a companion paper to \cite{JNT}. In \cite{JNT}, using ideas of Berend \cite{Bnumber} and Kra \cite{K}, it was proved that the sets of the form \{\lambda_1^n\mu_1^m\xi_1+\lambda_2^n\mu_2^m\xi_2:n,m\geq 1\}, where $\xi_1, \xi_2 \in \mathbb{R}$, $\lambda_1,\mu_1$ and $\lambda_2,\mu_2$ are two pairs of multiplicatively independent real algebraic numbers satisfying certain technical conditions, including that $\mu_i \in \mathbb{Q}(\lambda_i),$ $i=1,2,$ are dense modulo $1 \slash \kappa,$ for some $\kappa \geq 1$. In this paper we extend the result from \cite{JNT}, showing that the condition $\mu_i \in \mathbb{Q}(\lambda_i)$ can be removed by imposing appropriate conditions on the norms of conjugates of $\lambda_i,\mu_i$ and the degree of the algebraic numbers $\lambda_i^n \mu_i^m$.