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  • 1 Department of Mathematics, Faculty of Science Niigata University Ikarashi 2-8050, Niigata 950-2181, Japan Ikarashi 2-8050, Niigata 950-2181, Japan
  • 2 Haus-Endt-Straße 88, D-40593 Düsseldorf, Germany Haus-Endt-Straße 88, D-40593 Düsseldorf, Germany
  • 3 Faculty of Computer Science, University of Debrecen P.O. Box 12, H-4010 Debrecen, Hungary P.O. Box 12, H-4010 Debrecen, Hungary
  • 4 Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien Wiedner Hauptstraße 8-10, A-1040 Wien, Austria Wiedner Hauptstraße 8-10, A-1040 Wien, Austria
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Summary  

The periodicity of sequences of integers \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(a_{n})_{n\in\mathbb Z}$ \end{document} satisfying the inequalities

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$0 \le a_{n-1}+\lambda a_n +a_{n+1} < 1 \ (n \in {\mathbb Z})$$ \end{document}
is studied for real \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $ \lambda $ \end{document} with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $|\lambda|< 2$ \end{document}. Periodicity is proved in case \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $ \lambda $ \end{document} is the golden ratio; for other values of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $ \lambda $ \end{document} statements on possible period lengths are given. Further interesting results on the morphology of periods are illustrated. The problem is connected to the investigation of shift radix systems and of Salem numbers.