Subspace theorem

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In mathematics, the subspace theorem is a result obtained by Wolfgang M. Schmidt in 1972.[1] It states that if L1,...,Ln 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.

[edit] A corollary on Diophantine approximation

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

[edit] References

  1. ^ Schmidt, Wolfgang M. Norm form equations. Ann. of Math. (2) 96 (1972), pp. 526-551