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 Research reports

• Report number: 2003-012
• Authors: Christiane Frougny, Jean-Pierre Gazeau et Rudolf Krejcar
• Title: Additive and multiplicative properties of point setsbased on beta-integers
• Summary:

To each number $\beta>1$ correspond abelian groups in $\R^d$, of the form $\Lambda_\b=\sum_{i=1}^{d} \Z_\b {\bf e}_i$, which obey $\b \Lambda_\b \subset \Lambda_\b$. The set $\Z_{\beta}$ of beta-integers is a countable set of numbers : it is precisely the set of real numbers which are polynomial in $\beta$ when they are written in ''basis $\beta$'', and $\Z_{\beta} =\Z$ when $\beta \in \N$. We prove here a list of arithmetic properties of $\Z_{\beta}$: addition, multiplication, relation with integers, when $\beta$ is a quadratic Pisot-Vijayaraghavan unit (quasicrystallographic inflation factors are particular examples). We also consider the case of a cubic Pisot-Vijayaraghavan unit associated with the seven-fold cyclotomic ring. At the end, we show how the point sets $\Lambda_\beta$ are vertices of $d$-dimensional tilings.

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