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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Vojta's refinement of the subspace theorem


Author: Wolfgang M. Schmidt
Journal: Trans. Amer. Math. Soc. 340 (1993), 705-731
MSC: Primary 11J13; Secondary 11J61, 11J68
MathSciNet review: 1152325
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Abstract: Vojta's refinement of the Subspace Theorem says that given linearly independent linear forms $ {L_1}, \ldots , {L_n}$ in $ n$ variables with algebraic coefficients, there is a finite union $ U$ of proper subspaces of $ {\mathbb{Q}^n}$, such that for any $ \varepsilon > 0$ the points $ \underline{\underline x} \in {\mathbb{Z}^n}\backslash \{ \underline{\underline 0} \} $ with (1) $ \vert{L_1}(\underline{\underline x} ) \cdots {L_n}(\underline{\underline x} )\vert\; < \;\vert\underline{\underline x} {\vert^{ - \varepsilon }}$ lie in $ U$, with finitely many exceptions which will depend on $ \varepsilon $ . Put differently, if $ X(\varepsilon )$ is the set of solutions of (1), if $ \bar X(\varepsilon )$ is its closure in the subspace topology (whose closed sets are finite unions of subspaces) and if $ \bar X\prime(\varepsilon )$ consists of components of dimension $ > 1$ , then $ \bar X\prime(\varepsilon ) \subset U$ . In the present paper it is shown that $ \bar X\prime(\varepsilon )$ is in fact constant when $ \varepsilon $ lies outside a simply described finite set of rational numbers.

More generally, let $ k$ be an algebraic number field and $ S$ finite set of absolute values of $ k$ containing the archimedean ones. For $ \upsilon \in S$ let $ L_1^\upsilon, \ldots ,L_m^\upsilon$ be linear forms with coefficients in $ k$, and for $ \underline{\underline x} \in {K^n}\backslash \{ \underline{\underline 0} \} $ with height $ {H_k}(\underline{\underline x} ) > 1$ define $ {a_{\upsilon i}}(\underline{\underline x} )$ by $ \vert L_i^\upsilon(\underline{\underline x} )\vert _\upsilon/\vert\underline{\... ...{\underline x} )^{ - {a_{\upsilon i}}(\underline{\underline x} )/{d_\upsilon}}}$ where the $ {d_\upsilon}$ are the local degrees. The approximation set $ A$ consists of tuples $ \underline{\underline a} = \{ {a_{\upsilon i}}\} \;(\upsilon \in S,1 \leqq i \leqq m)$ such that for every neighborhood $ O$ of $ \underline{\underline a} $ the points $ \underline{\underline x} $ with $ \{ {a_{{v_i}}}\{ \underline{\underline x} )\} \in O$ are dense in the subspace topology. Then $ A$ is a polyhedron whose vertices are rational points.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1152325-3
PII: S 0002-9947(1993)1152325-3
Keywords: Simultaneous approximation to algebraic numbers, Subspace Theorem, subspace topology
Article copyright: © Copyright 1993 American Mathematical Society