Suites de longueur minimale associées à un ensemble normal donné. (Sequences of minimal length associated with a given normal set)
A set \({\mathcal A}\subseteq {\mathbb{R}}\) is called b-normal if there exists a bounded sequence \(\Lambda =(\lambda_ 1,\lambda_ 2,...)\) in \({\mathbb{R}}\) such that \({\mathcal A}\) is the set \({\mathcal B}(\Lambda)\) of all x for which \((x\lambda_ n)\) is u.d. mod 1. The author investigates the structure of such sets \({\mathcal A}\). In particular, he establishes relations between the ``minimal bound'' \[ M({\mathcal A}):=\inf \{M\geq 0:\quad \exists \Lambda,\quad {\mathcal A}={\mathcal B}(\Lambda),\quad 0\leq \lambda_ n\leq M\quad \forall n\} \] and various densities of \({\mathcal A}\). Especially interesting results are obtained in the case \({\mathcal A}\subseteq {\mathbb{Z}}\). E.g., any set \({\mathcal A}\) of the form \(\cup^{k}_{i=1}\gamma_ i({\mathbb{Z}}\setminus \{0\})\), with natural numbers \(\gamma_ i\), is shown to be b-normal and to satisfy \[ M({\mathcal A})\geq d({\mathcal A}):=\lim_{m\to \infty}(1/2m)\quad \#\{a\in {\mathcal A}:\quad | a| <m\}, \] with equality holding iff all coefficients of the polynomial \(P(t)=\prod_{a\in \bar A}(t-\zeta^ a_ m)\) are positive, where \(m=lcm\{\gamma_ 1,...,\gamma_ k\}\), \(\zeta_ m=e^{2\pi i/m}\), \(\bar A={\mathcal A}\cap \{1,2,...,m-1\}\).
scientific article; zbMATH DE number 4116647