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Mathematics of Computation

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Spectral properties of cubic complex Pisot units


Authors: Tomáš Hejda and Edita Pelantová
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
Published electronically: June 9, 2015
MathSciNet review: 3404455
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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)
Article copyright: © Copyright 2015 American Mathematical Society