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Toufik Zaimi
Commentaires sur quelques résultats sur les nombres de Pisot
Journal de théorie des nombres de Bordeaux, 22 no. 2 (2010), p. 513-524, doi: 10.5802/jtnb.729
Article PDF | Reviews MR 2769076 | Zbl 1223.11130

Résumé - Abstract

Comments on some results about Pisot numbers.

Using some results of Yves Meyer on harmonious sets, we prove that a real number $\theta >1$ is a Pisot number if and only if $A_{[\theta ]}\cup (-A_{[\theta ]}),$ where $A_{[\theta ]}$ is the set of polynomials with coefficients in $\lbrace 0,1,...,[\theta ]\rbrace $ evaluated at $\theta ,$ is a Meyer set. This allows us to deduce certain related results of Y. Bugeaud or P. Erdös and V. Komornik. By the same tools we also show that for $\varepsilon \in ]0,1],$ the set of $\varepsilon $-Pisot numbers which are contained in a real algebraic number field $K$ and have the same degree as $K,$ is a Meyer set.

Bibliography

[1] Y. Bugeaud, On a property of Pisot numbers and related questions. Acta Math. Hungar. 73 (1996), 33–39.  MR 1415918 |  Zbl 0923.11148
[2] P. Erdös, I. Joó and V. Komornik, Characterization of the unique expansion $1=\sum _{i\ge 1}q^{-n_{i}}$ and related problems. Bull. Soc. Math. France 118 (1990), 377–390. Numdam |  MR 1078082 |  Zbl 0721.11005
[3] P. Erdös and V. Komornik, Developments in non integer bases. Acta Math. Hungar. 79 (1998), 57–83.  MR 1611948 |  Zbl 0906.11008
[4] A. H. Fan and J. Schmeling, $\varepsilon $-Pisot numbers in any real algebraic number field are relatively dense. J. Algebra 272 (2004), 470–475.  MR 2028068 |  Zbl 1043.11074
[5] J. C. Lagarias, Meyer’s concept of quasicrystal and quasiregular sets. Commun. Math. Phys. 179 (1996), 365–376. Article |  MR 1400744 |  Zbl 0858.52010
[6] Y. Meyer, Algebraic numbers and harmonic analysis. North-Holland, 1972.  MR 485769 |  Zbl 0267.43001
[7] R. V. Moody, Meyer sets and their duals. The Mathematics of Long-Range Aperiodic Order, R. V. Moody, Ed., Kluwer 1997, 403–442.  MR 1460032 |  Zbl 0880.43008
[8] T. Zaïmi, On an approximation property of Pisot numbers II. J. Théor. Nombres Bordeaux 16 (2004), 239–249. Cedram |  MR 2145586 |  Zbl 1096.11037