On an approximation property of Pisot numbers II

Toufik Zaïmi[1]

  • [1] King Saud University Dept. of Mathematics P. O. Box 2455 Riyadh 11451, Saudi Arabia

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 1, page 239-249
  • ISSN: 1246-7405

Abstract

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Let q be a complex number, m be a positive rational integer and l m ( q ) = inf { P ( q ) , P ∈ ℤ m [ X ] , P ( q ) ≠ 0 } , where ℤ m [ X ] denotes the set of polynomials with rational integer coefficients of absolute value ≤ m . We determine in this note the maximum of the quantities l m ( q ) when q runs through the interval ] m , m + 1 [ . We also show that if q is a non-real number of modulus > 1 , then q is a complex Pisot number if and only if l m ( q ) > 0 for all m .

How to cite

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Zaïmi, Toufik. "On an approximation property of Pisot numbers II." Journal de Théorie des Nombres de Bordeaux 16.1 (2004): 239-249. <http://eudml.org/doc/249246>.

@article{Zaïmi2004,
abstract = {Let $q$ be a complex number, $m$ be a positive rational integer and $l_\{m\}(q)=\inf \lbrace \left| P(q)\right| ,P\in \mathbb\{Z\}_\{m\}[X],P(q)\ne 0\rbrace $, where $\mathbb\{Z\}_\{m\}[X]$ denotes the set of polynomials with rational integer coefficients of absolute value $\le m$. We determine in this note the maximum of the quantities $l_\{m\}(q)$ when $q$ runs through the interval $]m,m+1[$. We also show that if $q$ is a non-real number of modulus $&gt;1$, then $q$ is a complex Pisot number if and only if $l_\{m\}(q)&gt;0$ for all $m$.},
affiliation = {King Saud University Dept. of Mathematics P. O. Box 2455 Riyadh 11451, Saudi Arabia},
author = {Zaïmi, Toufik},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Pisot number; beta-number; spectra},
language = {eng},
number = {1},
pages = {239-249},
publisher = {Université Bordeaux 1},
title = {On an approximation property of Pisot numbers II},
url = {http://eudml.org/doc/249246},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Zaïmi, Toufik
TI - On an approximation property of Pisot numbers II
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 1
SP - 239
EP - 249
AB - Let $q$ be a complex number, $m$ be a positive rational integer and $l_{m}(q)=\inf \lbrace \left| P(q)\right| ,P\in \mathbb{Z}_{m}[X],P(q)\ne 0\rbrace $, where $\mathbb{Z}_{m}[X]$ denotes the set of polynomials with rational integer coefficients of absolute value $\le m$. We determine in this note the maximum of the quantities $l_{m}(q)$ when $q$ runs through the interval $]m,m+1[$. We also show that if $q$ is a non-real number of modulus $&gt;1$, then $q$ is a complex Pisot number if and only if $l_{m}(q)&gt;0$ for all $m$.
LA - eng
KW - Pisot number; beta-number; spectra
UR - http://eudml.org/doc/249246
ER -

References

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  11. V. Komornik, P. Loreti and M. Pedicini, An approximation property of Pisot numbers. J. Number Theory 80 (2000), 218–237. Zbl0962.11034MR1740512
  12. W. Parry, On the β - expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416. Zbl0099.28103MR142719
  13. A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Hungar. 8 (1957), 477–493. Zbl0079.08901MR97374
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  15. T. Zaïmi, On an approximation property of Pisot numbers. Acta Math. Hungar. 96 (4) (2002), 309–325. Zbl1012.11092MR1922677

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