Sequences of algebraic integers and density modulo $1$

Urban, Roman

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 3, pp. 755-762.

Nous établissons la densité modulo $1$ des ensembles de la forme

$$\{{\mu }^m{\lambda }^n\xi +r_m:n,m\in \mathbb{N}\},$$

où $\lambda ,\mu \in ℝ$ sont deux entiers algébriques de degré $d\ge 2,$ qui sont rationnellement indépendants et satisfont des hypothèses techniques supplémentaires, $\xi \ne 0,$ et $r_m$ une suite quelconque de nombres réels.

We prove density modulo $1$ of the sets of the form

$$\{{\mu }^m{\lambda }^n\xi +r_m:n,m\in \mathbb{N}\},$$

where $\lambda ,\mu \in ℝ$ is a pair of rationally independent algebraic integers of degree $d\ge 2,$ satisfying some additional assumptions, $\xi \ne 0,$ and $r_m$ is any sequence of real numbers.

Mots clés : Density modulo $1,$ algebraic integers, topological dynamics, ID-semigroups

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