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.

@article{JTNB_2004__16_1_197_0,
     author = {Werner Georg Nowak and Johannes Schoi\ss engeier},
     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 = {jtnb.centre-mersenne.org/item/JTNB_2004__16_1_197_0/}
}
Werner Georg Nowak; Johannes Schoißengeier. 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. https://jtnb.centre-mersenne.org/item/JTNB_2004__16_1_197_0/

[1] S. Bochner, Die Poissonsche Summenformel in mehreren Veränderlichen. Math. Ann. 106 (1932), 55–63. | MR 1512748 | Zbl 0003.25104

[2] N. Bourbaki, Elements of mathematics, Algebra II. Springer, Berlin 1990. | MR 1080964 | Zbl 0719.12001

[3] M.N. Huxley, Exponential sums and lattice points II. Proc. London Math. Soc. 66 (1993), 279-301. | MR 1199067 | Zbl 0820.11060

[4] M.N. Huxley, Area, lattice points, and exponential sums. LMS Monographs, New Ser. 13, Oxford 1996. | MR 1420620 | Zbl 0861.11002

[5] E. Krätzel, Lattice points. Kluwer, Dordrecht 1988. | MR 998378 | Zbl 0675.10031

[6] E. Krätzel, Analytische Funktionen in der Zahlentheorie. Teubner, Stuttgart 2000. | MR 1889901 | Zbl 0962.11001

[7] G. Kuba, On the distribution of squares of integral quaternions. Acta Arithm. 93 (2000), 359–372. | MR 1759481 | Zbl 0947.11028

[8] G. Kuba, On the distribution of squares of integral quaternions II. Acta Arithm. 101 (2002), 81–95. | MR 1879846 | Zbl 1004.11054

[9] G. Kuba, On the distribution of squares of hypercomplex integers. J. Number Th. 88 (2001), 313–334. | MR 1832009 | Zbl 0992.11056

[10] G. Kuba, Zur Verteilung der Quadrate ganzer Zahlen in rationalen Quaternionenalgebren. Abh. Math. Sem. Hamburg 72 (2002), 145–163. | MR 1941551 | Zbl 1014.11059

[11] G. Kuba, On the distribution of squares of integral Cayley numbers. Acta Arithm. 108 (2003), 253–265. | | MR 1980511 | Zbl 1028.11061

[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 1809556 | Zbl 1008.11039

[13] L. Kuipers and H. Niederreiter, Uniform distribution of sequences. J. Wiley, New York 1974. | MR 419394 | Zbl 0281.10001

[14] H. Müller and W.G. Nowak, Potenzen von Gaußschen ganzen Zahlen in Quadraten. Mitt. Math. Ges. Hamburg 18 (1999), 119–126. | MR 1727522 | Zbl 1027.11073

[15] W. Müller, On the average order of the lattice rest of a convex body. Acta Arithm. 80 (1997), 89–100. | | MR 1450420 | Zbl 0871.11070

[16] W. Narkiewicz, Elementary and analytic theory of algebraic numbers. 2 nd ed., Springer, Berlin 1990. | MR 1055830 | Zbl 02107000

[17] W.G. Nowak, Zur Verteilung der Potenzen Gaußscher ganzer Zahlen. Abh. Math. Sem. Hamburg 73 (2003), 43–65. | MR 2028506 | Zbl 1053.11081

[18] G. Opfer and W. Ripken, Complex version of Catalań’s problem. Mitt. Math. Ges. Hamburg 17 (1998), 101–112. | MR 1660928 | Zbl 1007.11014

[19] K.F. Roth, Rational approximations to algebraic numbers. Mathematika 2 (1955), 1–20. | MR 72182 | Zbl 0064.28501

[20] W.M. Schmidt, Simultaneous approximation to algebraic numbers by rationals. Acta Math. 125 (1970), 189–201. | MR 268129 | Zbl 0205.06702

[21] W.M. Schmidt, Diophantine approximation. LNM 785, Springer, Berlin 1980. | MR 568710 | Zbl 0421.10019

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