Spectral properties of cubic complex Pisot units

Abstract

For a real number $\beta >1$, Erdős, Joó and Komornik study distances between consecutive points in the set $X^m(\beta )=\{{\sum }_{j=0}^n{a}_j{\beta }^j:n\in \mathbb{N},a_j\in \{0,1,\dots ,m\}\}$. Pisot numbers play a crucial role for the properties of $X^m(\beta )$. Following the work of Zaïmi, who considered $X^m(\gamma )$ with $\gamma \in \mathbb{C}\setminus \mathbb{R}$ and $|\gamma |>1$, we show that for any non-real $\gamma$ and $m<|\gamma {|}^2-1$, the set $X^m(\gamma )$ is not relatively dense in the complex plane. Then we focus on complex Pisot units with a positive real conjugate ${\gamma }^{\prime }$ and $m>|\gamma {|}^2-1$. If the number $1/{\gamma }^{\prime }$ satisfies Property (F), we deduce that $X^m(\gamma )$ is uniformly discrete and relatively dense, i.e., $X^m(\gamma )$ is a Delone set. Moreover, we present an algorithm for determining two parameters of the Delone set $X^m(\gamma )$ which are analogous to minimal and maximal distances in the real case $X^m(\beta )$. For $\gamma$ satisfying ${\gamma }^3+{\gamma }^2+\gamma -1=0$, explicit formulas for the two parameters are given.

accepted to Math. Comp., 21 pages, 7 figures, 2 tables, 23 references