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 \[ X^m(\beta )=\Bigl \{\sum _{j=0}^n a_j \beta ^j : n\in \mathbb {N}, a_j\in \{0,1,\dots ,m\}\Bigr \}.\] 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 $\gamma$ with a positive real conjugate $\gamma ’$ and $m>|\gamma |^2-1$. If the number $1/\gamma ’$ 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.