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Algebraic numbers and density modulo $1,$ II

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Author(s): Roman Urban

Journal: Uniform Distribution Theory
ISSN 1336-913X

Volume: 5;
Issue: 1;
Start page: 111;
Date: 2010;
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Keywords: Density modulo 1 | algebraic numbers | multiplicatively independent numbers | topological dynamics | ID-semigroups | a-adic solenoid

ABSTRACT
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 $\lambda_i^n \mu_i^m$.
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