Spectral properties of cubic complex Pisot units

For a real number $\beta>1$, Erd\H{o}s, Jo\'o 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 Za\"imi, 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 $\gamma'$ 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.