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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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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