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

• Schmidt, Wolfgang M. (1972), "Norm form equations", Annals of Mathematics. Second Series, 96: 526–551, ISSN 0003-486X, JSTOR 1970824, MR 0314761

• Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
• Wolfgang M. Schmidt.Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000