Description of spectra of quadratic Pisot units

The spectrum of a real number β>1 is the set Xm(β) of p(β) where p ranges over all polynomials with coefficients restricted to A={0,1,…,m}. For a quadratic Pisot unit β, we determine the values of all distances between consecutive points and their corresponding frequencies, by recasting the spect...

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Bibliographic Details
Published in:Journal of number theory 2015-05, Vol.150, p.168-190
Main Authors: Masáková, Zuzana, Pastirčáková, Kateřina, Pelantová, Edita
Format: Article
Language:English
Subjects:
Interval exchange
Pisot numbers
Spectrum
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Summary:The spectrum of a real number β>1 is the set Xm(β) of p(β) where p ranges over all polynomials with coefficients restricted to A={0,1,…,m}. For a quadratic Pisot unit β, we determine the values of all distances between consecutive points and their corresponding frequencies, by recasting the spectra in the frame of the cut-and-project scheme. We also show that shifting the set A of digits so that it contains at least one negative element, or considering negative base −β instead of β, the gap sequence of the modified spectrum is a coding of an exchange of three intervals.
ISSN:0022-314X
1096-1658
DOI:10.1016/j.jnt.2014.11.011