Spectral Properties of Cubic Complex Pisot Units

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F16%3A00235680" target="_blank" >RIV/68407700:21340/16:00235680 - isvavai.cz</a>
Result on the web
<a href="http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02983-4/" target="_blank" >http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02983-4/</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/mcom/2983" target="_blank" >10.1090/mcom/2983</a>

Result language
angličtina
Original language name
Spectral Properties of Cubic Complex Pisot Units
Original language description
For a real number $beta>1$, ErdH{o}s, Jo'o and Komornik study distances between consecutive points in the set $X^m(beta)=bigl{sum_{j=0}^n a_j beta^j : ninmathbb N,,a_jin{0,1,dots,m}bigr}$. 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 $gammainmathbb{C}setminusmathbb{R}$ 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.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/GA13-03538S" target="_blank" >GA13-03538S: Algorithms, Dynamics and Geometry of Numeration systems</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2016
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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