Schmidt's subspace theorem with moving targets
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 condition (ii) in the Theorem is not satisfied, is discussed in Sect. 3. Wirsing's theorem with moving targets, which generalizes Roth's theorem with moving targets, is derived in Sect. 4. See also the first author's general form of the second main theorem [Trans. Am. Math. Soc. 349, 5093-5105 (1999; Zbl 0891.32001)].
scientific article; zbMATH DE number 1026495