Algebraic Numbers, Hyperbolicity, and Density Modulo One

We prove the density of the sets of the form ${λ_1^m μ_1^n ξ_1 +...+λ_k^m μ_k^n ξ_k : m,n \in \mathbb N}$ modulo one, where $λ_i$ and $μ_i$ are multiplicatively independent algebraic numbers satisfying some additional assumptions. The proof is based on analysing dynamics of higher-rank actions on compact abelean groups.

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