Algebraic Numbers, Hyperbolicity, and Density Modulo One

Abstract

We prove the density of the sets of the form ${\lambda }_1^m{\mu }_1^n{\xi }_1+...+{\lambda }_k^m{\mu }_k^n{\xi }_k:m,n\in \mathbb{N}$ modulo one, where ${\lambda }_i$ and ${\mu }_i$ are multiplicatively independent algebraic numbers satisfying some additional assumptions. The proof is based on analysing dynamics of higher-rank actions on compact abelean groups.