The subspace theorem 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

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

lie in a finite number of proper subspaces of Qⁿ.

A quantitative form of the theorem, which determines the number of subspaces containing all solutions, was also obtained by Schmidt, and the theorem was generalised by Schlickewei (1977) to allow more general absolute values on number fields.

The theorem may be used to obtain results on Diophantine equations such as Siegel's theorem on integral points and solution of the S-unit equation.^[1]

A corollary on Diophantine approximation edit

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

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

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.

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• Schlickewei, Hans Peter (1977). "On norm form equations". J. Number Theory. 9 (3): 370–380. doi:10.1016/0022-314X(77)90072-5. MR 0444562.
• Schmidt, Wolfgang M. (1972). "Norm form equations". Annals of Mathematics. Second Series. 96 (3): 526–551. doi:10.2307/1970824. JSTOR 1970824. MR 0314761.
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