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Effective diophantine approximation on \(\mathbb{G}_M\). II. (English) Zbl 0912.11028

In a 1993 paper [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 20, 61-89 (1993; Zbl 0774.11034)], E. Bombieri introduced, in the archimedean case, a new method for obtaining effective rationality measures for roots of high order of algebraic numbers and examined the applications to effective diophantine approximation in a number field by a finitely generated subgroup. A new effective solution of Thue’s equation in number fields and the Baker-Feldman theorem resulted. The main tools were the Thue-Siegel Principle, Viola’s version of Dyson’s Lemma, and the geometry of numbers, there being no appeal to linear forms in logarithms. Here the authors extend this work to the non-archimedean case and introduce, along the lines of ideas by P. Corvaja [Ann. Inst. Fourier 45, 1177-1203 (1995; Zbl 0833.11030) and Monatsh. Math. 124, No. 2, 147-175 (1997; Zbl 0883.11033)], the use of Laurent’s determinantal method to replace Siegel’s Lemma in this context. Here is Theorem 1 of the paper under review.
Let \(K\) be a number field of degree \(d\) and \(a\in K\) not equal to zero or a root of unity. Let \(p\) be a rational prime and suppose that \(v\) is a place of \(K\) dividing \(p\) such that \(|\alpha-1|_v<1\). Let \(r\) be a positive integer coprime with \(p\). Then \(a\) has an \(r\)-th root \(\alpha\) satisfying \(0<|\alpha-1| _{\widetilde{v}}<1\) for a place \(\widetilde{v}\) of \(K(\alpha)\) extending \(v\), where \(|\;| _{\widetilde{v}}\) is normalized so as to agree with \(|\;| _{v}\) on \(K\). Let \(h(\;)\) denote Weil’s absolute logarithmic height, \(f_{v}\) the residue class degree, \[ d_{\nu}^{*}={d\over f_{v}\log p} \quad \text{and} \quad D_{\nu}^{*}=\max\{1,d_{\nu}^{*}\}. \] Let \(\kappa\) be a real number in the range \(0<\kappa\leq 1\). Define \(c_{1}=3.4\cdot 10^{11}(D_{\nu}^{*})^{10}{1\over\kappa} (\log{1\over\kappa}+1)^{7}\) and \(c_{2}= 1.7\cdot 10^{6}(D_{\nu}^{*})^{4}{1\over\kappa} (\log{1\over\kappa}+1)^{4}\). Let \(\alpha'=\alpha\gamma\) with \(\gamma\in K\), \(\gamma\not=0\). Then the inequalities \[ r\geq c_{1} h(\alpha) \quad \text{and} \quad h(\alpha')\geq c_{2} \] imply \[ |\alpha'-1|_{\widetilde{v}}\geq H(\alpha')^{-r\kappa}. \] The authors apply this result to diophantine approximation in a number field by a finitely generated multiplicative subgroup; their Theorem 2 reads:
Let \(\Gamma\) be a finitely generated subgroup of \(K^{*}\) and let \(\xi_{1},\ldots,\xi_{t}\) be generators of \(\Gamma\) modulo torsion. Let \(\xi\in\Gamma\), \(A\in K^{*}\) and \(\kappa>0\) be such that \(0<| 1-A\xi| _{v}<H(A\xi)^{-\kappa}\leq 1\). Define \(h'(\xi_{i})=\max\{h(\xi_{i}),1/d_v^{*}\}\) and \[ Q=(2c_{1}t)^{t}(50c_{2})p^{f_{v}}\prod_{i=1}^{t}h'(\xi_{= i}). \] Then we have \[ h(A\xi)\leq c_{1}Q\max\{h'(A),Q\}. \] The deduction of Theorem 2 from Theorem 1 can be carried out also in the case of archimedean valuations and yields an improvement of Theorem 2 of Bombieri’s earlier paper (op. cit.).
It is stated in this paper that Theorem 1 can be obtained directly from Baker’s method, rather than from the Thue-Siegel method. The authors point out that this would lead to a sharper version of their Theorem 1 and that their Theorem 2, whose proof uses the geometry of numbers, could then be applied directly to this sharper result. This program has been carried out by Yann Bugeaud, both in the archimedean and non-archimedean cases [Y. Bugeaud, Bornes effectives pour les solutions des équations en \(S\)-unités et des équations de Thue-Mahler, J. Number Theory 71, No. 2, 227-244 (1998)].

MSC:

11J82 Measures of irrationality and of transcendence
11J61 Approximation in non-Archimedean valuations
14G05 Rational points
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References:

[1] A. Baker - G. Wüstholz , Logarithmic forms and group varieties , J. reine angew. Math . 442 ( 1993 ), 19 - 62 . Zbl 0788.11026 · Zbl 0788.11026 · doi:10.1515/crll.1993.442.19
[2] E. Bombieri , Effective Diophantine Approximation on G m , Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20 ( 1993 ), 61 - 89 . Numdam | Zbl 0774.11034 · Zbl 0774.11034
[3] E. Bombieri - J. Vaaler , On Siegel’s lemma , Invent. Math. 73 ( 1983 ), 11 - 32 ; Addendum , ibid 75 ( 1984 ), 177 . Zbl 0533.10030 · Zbl 0533.10030 · doi:10.1007/BF01393823
[4] T. Struppeck - J.D. Vaaler , Inequalities for heights of algebraic subspaces and the Thue-Siegel Principle, Analytic Number Theory, Proceedings of a Conference in Honor of Paul T . Bateman, B.C. Berndt, H.G. Diamond, H. Halberstam, A. Hildebrand eds., Progr. Math . Birkhäuser , Boston 85 ( 1990 ), 493 - 528 . Zbl 0722.11033 · Zbl 0722.11033
[5] P. Corvaja , Autour du théorème de Roth , preprint. · Zbl 0883.11033
[6] P. Corvaja , Une application nouvelle de la méthode de Thue , Ann. Inst. Fourier , Grenoble 45 ( 1995 ), 1177 - 1203 . Numdam | Zbl 0833.11030 · Zbl 0833.11030 · doi:10.5802/aif.1491
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