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

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.

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.

DOI: 10.5802/jtnb.442
Werner Georg Nowak 1; Johannes Schoißengeier 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
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Werner Georg Nowak; Johannes Schoißengeier. The distribution of powers of integers in algebraic number fields. Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 1, pp. 197-214. doi : 10.5802/jtnb.442. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.442/

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