## Spectral properties of cubic complex Pisot units

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- by Tomáš Hejda and Edita Pelantová PDF
- Math. Comp.
**85**(2016), 401-421 Request permission

## 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.

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## Additional Information

**Tomáš Hejda**- Affiliation: Department of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, Prague 12000, Czech Republic
- Address at time of publication: LIAFA, CNRS UMR 7089, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France
- Email: tohecz@gmail.com
**Edita Pelantová**- Affiliation: Department of Mathematics FNSPE, Czech Technical University in Prague, Trojanova 13, Prague 12000, Czech Republic
- Email: edita.pelantova@fjfi.cvut.cz
- Received by editor(s): December 2, 2013
- Received by editor(s) in revised form: May 14, 2014, and August 5, 2014
- Published electronically: June 9, 2015
- Additional Notes: This work was supported by Grant Agency of the Czech Technical University in Prague grant SGS14/205/OHK4/3T/14, Czech Science Foundation grant 13-03538S, and ANR/FWF project “FAN – Fractals and Numeration” (ANR-12-IS01-0002, FWF grant I1136)
- © Copyright 2015 American Mathematical Society
- Journal: Math. Comp.
**85**(2016), 401-421 - MSC (2010): Primary 11A63, 11K16, 52C23, 52C10; Secondary 11H99, 11-04
- DOI: https://doi.org/10.1090/mcom/2983
- MathSciNet review: 3404455