On an approximation property of Pisot numbers II
Journal de Théorie des Nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 239-249.

Soit q un nombre complexe, m un entier positif et l m (q)=inf{P(q),P m [X],P(q)0}, où m [X] désigne l’ensemble des polynômes à coefficients entiers de valeur absolue m. Nous déterminons dans cette note le maximum des quantités l m (q) quand q décrit l’intervalle ]m,m+1[. Nous montrons aussi que si q est un nombre non-réel de module >1, alors q est un nombre de Pisot complexe si et seulement si l m (q)>0 pour tout m.

Let q be a complex number, m be a positive rational integer and l m (q)=inf{P(q),P m [X],P(q)0}, where 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.

@article{JTNB_2004__16_1_239_0,
     author = {Toufik Za\"\i mi},
     title = {On an approximation property of Pisot numbers II},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {239--249},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {1},
     year = {2004},
     doi = {10.5802/jtnb.446},
     zbl = {02184644},
     mrnumber = {2145586},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2004__16_1_239_0/}
}
Toufik Zaïmi. On an approximation property of Pisot numbers II. Journal de Théorie des Nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 239-249. doi : 10.5802/jtnb.446. https://jtnb.centre-mersenne.org/item/JTNB_2004__16_1_239_0/

[1] K. Alshalan and T. Zaimi, Some computations on the spectra of Pisot numbers. Submitted.

[2] D. Berend and C. Frougny, Computability by finite automata and Pisot Bases. Math. Systems Theory 27 (1994), 275–282. | MR 1264390 | Zbl 0819.11005

[3] P. Borwein and K. G. Hare, Some computations on the spectra of Pisot and Salem numbers. Math. Comp. 71 No. 238 (2002), 767–780. | MR 1885627 | Zbl 1037.11065

[4] D. W. Boyd, Salem numbers of degree four have periodic expansions. Number Theory (eds J.-H. de Coninck and C. Levesque, Walter de Gruyter, Berlin) 1989, 57–64. | MR 1024551 | Zbl 0685.12004

[5] Y. Bugeaud, On a property of Pisot numbers and related questions. Acta Math. Hungar. 73 (1996), 33–39. | MR 1415918 | Zbl 0923.11148

[6] P. Erdös, I. Joó and V. Komornik, Characterization of the unique expansions 1= i1 q -n i and related problems. Bull. Soc. Math. France 118 (1990), 377–390. | Numdam | MR 1078082 | Zbl 0721.11005

[7] P. Erdös, I. Joó and V. Komornik, On the sequence of numbers of the form ε 0 +ε 1 q+...+ε n q n ε i {0,1}. Acta Arith. 83 (1998), 201–210. | MR 1611185 | Zbl 0896.11006

[8] P. Erdös, I. Joó and F. J. Schnitzer, On Pisot numbers. Ann. Univ. Sci. Budapest Eotvos Sect. Math. 39 (1996), 95–99. | MR 1451448 | Zbl 0880.11067

[9] P. Erdös and V. Komornik, Developments in non integer bases. Acta Math. Hungar. 79 (1998), 57–83. | MR 1611948 | Zbl 0906.11008

[10] C. Frougny, Representations of numbers and finite automata. Math. Systems Theory 25 (1992), 37–60. | MR 1139094 | Zbl 0776.11005

[11] V. Komornik, P. Loreti and M. Pedicini, An approximation property of Pisot numbers. J. Number Theory 80 (2000), 218–237. | MR 1740512 | Zbl 0962.11034

[12] W. Parry, On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416. | MR 142719 | Zbl 0099.28103

[13] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Hungar. 8 (1957), 477–493. | MR 97374 | Zbl 0079.08901

[14] B. Solomyak, Conjugates of beta-numbers and the zero-free domain for a class of analytic functions. Proc. London Math. Soc. 68 (1994), 477–498. | MR 1262305 | Zbl 0820.30007

[15] T. Zaïmi, On an approximation property of Pisot numbers. Acta Math. Hungar. 96 (4) (2002), 309–325. | MR 1922677 | Zbl 1012.11092