The distribution of powers of integers in algebraic number fields
Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 197-214.

Pour tout corps de nombres K (non totalement réel), se pose la question de déterminer le nombre de puissances p-ièmes d’entiers algébriques γ de K, vérifiant μ(τ(γ p ))X, ceci pour tout plongement τ de K dans le corps des nombres complexes. Ici, X est un paramètre réel grand, p est un entier fixé 2 et μ(z)=max(| Re (z)|,| Im (z)|) (z, nombre complexe). Ce nombre est évalué asymptotiquement sous la forme c p,K X n/p +R p,K (X), avec des estimations précises sur le reste R p,K (X). La démonstration utilise des techniques issues de la théorie des réseaux, dont en particulier la généralisation multidimensionnelle, donnée par W. Schmidt, du théorème de K.F. Roth sur l’approximation des nombres algébriques par les nombres rationnels.

For an arbitrary (not totally real) number field K of degree 3, we ask how many perfect powers γ p of algebraic integers γ in K exist, such that μ(τ(γ p ))X for each embedding τ of K into the complex field. (X a large real parameter, p2 a fixed integer, and μ(z)=max(| Re (z)|,| Im (z)|) for any complex z.) This quantity is evaluated asymptotically in the form c p,K X n/p +R p,K (X), with sharp estimates for the remainder R p,K (X). The argument uses techniques from lattice point theory along with W. Schmidt’s multivariate extension of K.F. Roth’s result on the approximation of algebraic numbers by rationals.

DOI : 10.5802/jtnb.442
Nowak, Werner Georg 1 ; Schoißengeier, Johannes 2

1 Institute of Mathematics Department of Integrative Biology BOKU - University of Natural Resources and Applied Life Sciences Peter Jordan-Straße 82 A-1190 Wien, Austria
2 Institut für Mathematik der Universität Wien Nordbergstraße 15 A-1090 Wien, Austria
@article{JTNB_2004__16_1_197_0,
     author = {Nowak, Werner Georg and Schoi{\ss}engeier, Johannes},
     title = {The distribution of powers of integers in algebraic number fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {197--214},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {1},
     year = {2004},
     doi = {10.5802/jtnb.442},
     zbl = {1079.11057},
     mrnumber = {2145581},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jtnb.442/}
}
TY  - JOUR
AU  - Nowak, Werner Georg
AU  - Schoißengeier, Johannes
TI  - The distribution of powers of integers in algebraic number fields
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2004
SP  - 197
EP  - 214
VL  - 16
IS  - 1
PB  - Université Bordeaux 1
UR  - http://www.numdam.org/articles/10.5802/jtnb.442/
DO  - 10.5802/jtnb.442
LA  - en
ID  - JTNB_2004__16_1_197_0
ER  - 
%0 Journal Article
%A Nowak, Werner Georg
%A Schoißengeier, Johannes
%T The distribution of powers of integers in algebraic number fields
%J Journal de théorie des nombres de Bordeaux
%D 2004
%P 197-214
%V 16
%N 1
%I Université Bordeaux 1
%U http://www.numdam.org/articles/10.5802/jtnb.442/
%R 10.5802/jtnb.442
%G en
%F JTNB_2004__16_1_197_0
Nowak, Werner Georg; Schoißengeier, Johannes. The distribution of powers of integers in algebraic number fields. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 197-214. doi : 10.5802/jtnb.442. http://www.numdam.org/articles/10.5802/jtnb.442/

[1] S. Bochner, Die Poissonsche Summenformel in mehreren Veränderlichen. Math. Ann. 106 (1932), 55–63. | MR | Zbl

[2] N. Bourbaki, Elements of mathematics, Algebra II. Springer, Berlin 1990. | MR | Zbl

[3] M.N. Huxley, Exponential sums and lattice points II. Proc. London Math. Soc. 66 (1993), 279-301. | MR | Zbl

[4] M.N. Huxley, Area, lattice points, and exponential sums. LMS Monographs, New Ser. 13, Oxford 1996. | MR | Zbl

[5] E. Krätzel, Lattice points. Kluwer, Dordrecht 1988. | MR | Zbl

[6] E. Krätzel, Analytische Funktionen in der Zahlentheorie. Teubner, Stuttgart 2000. | MR | Zbl

[7] G. Kuba, On the distribution of squares of integral quaternions. Acta Arithm. 93 (2000), 359–372. | MR | Zbl

[8] G. Kuba, On the distribution of squares of integral quaternions II. Acta Arithm. 101 (2002), 81–95. | MR | Zbl

[9] G. Kuba, On the distribution of squares of hypercomplex integers. J. Number Th. 88 (2001), 313–334. | MR | Zbl

[10] G. Kuba, Zur Verteilung der Quadrate ganzer Zahlen in rationalen Quaternionenalgebren. Abh. Math. Sem. Hamburg 72 (2002), 145–163. | MR | Zbl

[11] G. Kuba, On the distribution of squares of integral Cayley numbers. Acta Arithm. 108 (2003), 253–265. | EuDML | MR | Zbl

[12] G. Kuba, H. Müller, W.G. Nowak and J. Schoißengeier, Zur Verteilung der Potenzen ganzer Zahlen eines komplexen kubischen Zahlkörpers. Abh. Math. Sem. Hamburg 70 (2000), 341–354. | MR | Zbl

[13] L. Kuipers and H. Niederreiter, Uniform distribution of sequences. J. Wiley, New York 1974. | MR | Zbl

[14] H. Müller and W.G. Nowak, Potenzen von Gaußschen ganzen Zahlen in Quadraten. Mitt. Math. Ges. Hamburg 18 (1999), 119–126. | MR | Zbl

[15] W. Müller, On the average order of the lattice rest of a convex body. Acta Arithm. 80 (1997), 89–100. | EuDML | MR | Zbl

[16] W. Narkiewicz, Elementary and analytic theory of algebraic numbers. 2 nd ed., Springer, Berlin 1990. | MR | Zbl

[17] W.G. Nowak, Zur Verteilung der Potenzen Gaußscher ganzer Zahlen. Abh. Math. Sem. Hamburg 73 (2003), 43–65. | MR | Zbl

[18] G. Opfer and W. Ripken, Complex version of Catalań’s problem. Mitt. Math. Ges. Hamburg 17 (1998), 101–112. | MR | Zbl

[19] K.F. Roth, Rational approximations to algebraic numbers. Mathematika 2 (1955), 1–20. | MR | Zbl

[20] W.M. Schmidt, Simultaneous approximation to algebraic numbers by rationals. Acta Math. 125 (1970), 189–201. | MR | Zbl

[21] W.M. Schmidt, Diophantine approximation. LNM 785, Springer, Berlin 1980. | MR | Zbl

[22] Wolfram Research, Inc. Mathematica, Version 4.0. Wolfram Research, Inc. Champaign 1999.

Cité par Sources :