Schmidt's subspace theorem with moving targets
type: Item
instance of: scholarly article
title: Schmidt's subspace theorem with moving targets
zbMATH Open document ID: 1013.11044
author: Min Ru
author: Paul Vojta
published in: Inventiones Mathematicae
publication date: 1997-12-08T00:00:00Z
review text: The second main theorem for holomorphic curves in Nevanlinna theory, via Vojta's dictionary, is Schmidt's subspace theorem. In 1991, \textit{Min Ru} and \textit{W. Stoll} [J. Geom. Anal. 1, 99-138 (1991; Zbl 0732.30025)] obtained the second main theorem with slowly moving targets for holomorphic curves \(f : \mathbb{C}\to \mathbb{P}^n\). The purpose of this paper is to prove the counterpart of this theorem in number theory: Schmidt's subspace theorem with moving targets. The method of proof is a combination of the proof of the second main theorem with moving targets for holomorphic curves [Ru-Stoll (loc. cit.)] and the proof of Roth's theorem with moving targets in \textit{P. Vojta} [Math. Res. Not. 1996, 109-114 (1996; Zbl 0877.11041)]. It is worth noting that this paper basically says that ``Schmidt's subspace theorem (with fixed targets) implies Schmidt's subspace theorem with moving targets''. Theorem. (Schmidt's subspace theorem with moving targets). Let \(k\) be a number field, let \(S\) be a finite set of places of \(k\), let \(q\in\mathbb{Z}_{>0}\), and let \(\varepsilon> 0\). Let \(\Lambda\) be an infinite index set, let \(H_1,\dots,H_q\) be moving hyperplanes \(\Lambda\to (\mathbb{P}^n)^*(k)\), and let \({\mathbf x} : \Lambda\to \mathbb{P}^n(k)\) be a collection of points such that: (i) for each \(\alpha\in \Lambda\), \(H_1(\alpha),\dots,H_q(\alpha)\) are in general position; (ii) \(\mathbf x\) is non-degenerate over \(\mathcal R\) (defined in the paper); and (iii) \(h(H_j(\alpha)) =o(h({\mathbf x}(\alpha)))\) for all \(j =1,\dots,q\). Then there exists an infinite index subset \(A\subseteq \Lambda\) such that \[ \frac{1}{[k:\mathbb{Q}]} \sum_{v\in S}\sum_{j=1}^q\lambda_{H_j(\alpha),v}({\mathbf x}(\alpha))\leq (n+1+\varepsilon)h({\mathbf x}(\alpha)) \] for all \(\alpha\in A\). If \(n = 1\), the non-degeneracy condition is automatically satisfied if \(h(H_j(\alpha)) =o(h(x(\alpha)))\), \(j=1,\dots,q\). The case of ``degeneracy'', i.e. the case when conditio…
Mathematics Subject Classification ID: 11J97
Mathematics Subject Classification ID: 32H30
zbMATH DE Number: 1026495
reviewed by: Olaf Ninnemann
MaRDI profile type: Publication
full work available at URL: s002220050114
OpenAlex ID: W2056654666
DOI: 10.1007/S002220050114
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description: scientific article; zbMATH DE number 1026495