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

DOI:
https://doi.org/10.1090/S0002-9947-1993-1152325-3

MathSciNet review:
1152325

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Abstract: Vojta's refinement of the Subspace Theorem says that given linearly independent linear forms in variables with algebraic coefficients, there is a finite union of proper subspaces of , such that for any the points with (1) lie in , with finitely many exceptions which will depend on . Put differently, if is the set of solutions of (1), if is its closure in the subspace topology (whose closed sets are finite unions of subspaces) and if consists of components of dimension , then . In the present paper it is shown that is in fact constant when lies outside a simply described finite set of rational numbers.

More generally, let be an algebraic number field and finite set of absolute values of containing the archimedean ones. For let be linear forms with coefficients in , and for with height define by where the are the local degrees. The approximation set consists of tuples such that for every neighborhood of the points with are dense in the subspace topology. Then is a polyhedron whose vertices are rational points.

**[1]**Hans Peter Schlickewei,*Die 𝑝-adische Verallgemeinerung des Satzes von Thue-Siegel-Roth-Schmidt*, J. Reine Angew. Math.**288**(1976), 86–105 (German). MR**0422166**, https://doi.org/10.1515/crll.1976.288.86**[2]**Wolfgang M. Schmidt,*Norm form equations*, Ann. of Math. (2)**96**(1972), 526–551. MR**0314761**, https://doi.org/10.2307/1970824**[3]**Wolfgang M. Schmidt,*Diophantine approximation*, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR**568710****[4]**J. H. Silverman,*The arithmetic of elliptic curves*, Graduate Texts in Math., Springer, 1985.**[5]**Paul Vojta,*A refinement of Schmidt’s subspace theorem*, Amer. J. Math.**111**(1989), no. 3, 489–518. MR**1002010**, https://doi.org/10.2307/2374670

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1152325-3

Keywords:
Simultaneous approximation to algebraic numbers,
Subspace Theorem,
subspace topology

Article copyright:
© Copyright 1993
American Mathematical Society