# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Spectral properties of cubic complex Pisot unitsHTML articles powered by AMS MathViewer

by Tomáš Hejda and Edita Pelantová
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.

References
Similar Articles
• 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)