Subspace theorem

In mathematics, the subspace theorem is a result obtained by Wolfgang M. Schmidt (1972). It states that if L₁,...,L_n are linearly independent linear forms in n variables with algebraic coefficients and if ε>0 is any given real number, then the non-zero integer points x with

$|L_{1}(x)\cdots L_{n}(x)|<|x|^{-\epsilon }$

lie in a finite number of proper subspaces of Qⁿ.

Schmidt's subspace theorem was generalised in by Schlickewei (1977) to allow more general absolute values.

A corollary on Diophantine approximation

The following corollary to the subspace theorem is often itself referred to as the subspace theorem. If a₁,...,a_n are algebraic such that 1,a₁,...,a_n are linearly independent over Q and ε>0 is any given real number, then there are only finitely many rational n-tuples (x₁/y,...,x_n/y) with

$|a_{i}-x_{i}/y|<y^{-(1+1/n+\epsilon )},\quad i=1,\ldots ,n.$

The specialization n = 1 gives the Thue–Siegel–Roth theorem. One may also note that the exponent 1+1/n+ε is best possible by Dirichlet's theorem on diophantine approximation.

References

• Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. Cambridge University Press. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034.
• Schmidt, Wolfgang M. (1972), "Norm form equations", Annals of Mathematics. Second Series, 96: 526–551, ISSN 0003-486X, JSTOR 1970824, MR 0314761
• Schmidt, Wolfgang M. (1980). Diophantine approximation. Lecture Notes in Mathematics. Vol. 785 (1996 with minor corrections ed.). Springer-Verlag.
• Schmidt, Wolfgang M. (1996). Diophantine approximations and Diophantine equations. Lecture Notes in Mathematics. Vol. 1467 (2nd ed ed.). Springer-Verlag. ISBN 3-540-54058-X. Zbl 0754.11020. {{cite book}}: |edition= has extra text (help)