Comptes Rendus
Number Theory
A property of the spectra of non-Pisot numbers
Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 121-124.

Let θ be a real number satisfying 1<θ<2, m a positive rational integer and Bm(θ) the set of polynomials with coefficients in {0,±1,,±m}, evaluated at θ. We prove that Bm(θ) is everywhere dense when 0Bm(θ), where Bm(θ) is the derivative set of Bm(θ). We also show that if Bm(θ)[0, 1θk0(11θ2k)]=, then Bm(θ) is discrete.

Soient θ un nombre réel satisfaisant 1<θ<2, m un entier rationnel positif et Bm(θ) l'ensemble des réels P(θ) pour P décrivant les polynômes à coefficients dans {0,±1,,±m}. On montre que Bm(θ) est partout dense lorsque 0 est un élément de l'ensemble dérivé Bm(θ) de Bm(θ). On prouve également que si Bm(θ)[0, 1θk0(11θ2k)]=, alors Bm(θ) est discret.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2010.01.016
Toufik Zaïmi 1

1 Département de mathématiques, Université Larbi Ben M'hidi, Oum El Bouaghi 04000, Algeria
@article{CRMATH_2010__348_3-4_121_0,
     author = {Toufik Za{\"\i}mi},
     title = {A property of the spectra of {non-Pisot} numbers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {121--124},
     publisher = {Elsevier},
     volume = {348},
     number = {3-4},
     year = {2010},
     doi = {10.1016/j.crma.2010.01.016},
     language = {en},
}
TY  - JOUR
AU  - Toufik Zaïmi
TI  - A property of the spectra of non-Pisot numbers
JO  - Comptes Rendus. Mathématique
PY  - 2010
SP  - 121
EP  - 124
VL  - 348
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crma.2010.01.016
LA  - en
ID  - CRMATH_2010__348_3-4_121_0
ER  - 
%0 Journal Article
%A Toufik Zaïmi
%T A property of the spectra of non-Pisot numbers
%J Comptes Rendus. Mathématique
%D 2010
%P 121-124
%V 348
%N 3-4
%I Elsevier
%R 10.1016/j.crma.2010.01.016
%G en
%F CRMATH_2010__348_3-4_121_0
Toufik Zaïmi. A property of the spectra of non-Pisot numbers. Comptes Rendus. Mathématique, Volume 348 (2010) no. 3-4, pp. 121-124. doi : 10.1016/j.crma.2010.01.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.016/

[1] P. Borwein; K.G. Hare Some computations on the spectra of Pisot and Salem numbers, Math. Comp., Volume 71 (2002), pp. 767-780

[2] Y. Bugeaud On a property of Pisot numbers and related questions, Acta Math. Hungar., Volume 73 (1996), pp. 33-39

[3] P. Erdős; I. Joó; V. Komornik Characterization of the unique expansion 1=i1qni and related problems, Bull. Soc. Math. France, Volume 118 (1990), pp. 377-390

[4] P. Erdős; I. Joó; F.J. Schnitzer On Pisot numbers, Ann. Univ. Sci. Budapest, Volume 39 (1996), pp. 95-99

[5] P. Erdős; V. Komornik Developments in non-integer bases, Acta Math. Hungar., Volume 79 (1998), pp. 57-83

[6] T. Zaïmi Approximation by polynomials with bounded coefficients, J. Number Theory, Volume 127 (2007), pp. 103-117

[7] T. Zaïmi Une remarque sur le spectre des nombres de Pisot, C. R. Acad. Sci. Paris Ser. I, Volume 347 (2009), pp. 5-8

Cited by Sources:

Comments - Policy


Articles of potential interest

Une remarque sur le spectre des nombres de Pisot

Toufik Zaïmi

C. R. Math (2009)


Complex Pisot numbers in algebraic number fields

Marie José Bertin; Toufik Zaïmi

C. R. Math (2015)