VOJTA REFINEMENT OF THE SUBSPACE THEOREM

Vojta's refinement of the Subspace Theorem says that given linearly in dependent linear forms L1,..., L(n) in n variables with algebraic coef ficients, there is a finite union U of proper subspaces of Q(n) , such that for any epsilon > 0 the points x is-an-element-of Z(n)\{0} with (1) \L1(x)...L(n)(x)\ < Absolute value of x -epsilon lie in U, with fi nitely many exceptions which will depend on epsilon. Put differently, if X(epsilon) is the set of solutions of (1), if XBAR(epsilon) is its closure in the subspace topology (whose closed sets are finite unions of subspaces) and if XBAR'(epsilon) consists of components of dimensio n > 1 , then XBAR'(epsilon) subset-of U . In the present paper it is s hown that XBAR'(epsilon) is in fact constant when epsilon lies outside a simply described finite set of rational numbers. More generally, le t k be an algebraic number field and S a finite set of absolute values of k containing the archimedean ones. For v is-an-element-of S let L1 v,..., L(m)v be linear forms with coefficients in k , and for x is-an- element-of K(n)\{0} with height H(k)(x) > 1 define a(vi)(x) by \L(i)v( x)\v/Absolute value of x v = H(k)(x)-a(vi)(x)/d(v) where the d(v) are the local degrees. The approximation set A consists of tuples a = {a(v i)} (v is-an-element-of S, 1 less-than-or-equal-to i less-than-or-equa l-to m) such that for every neighborhood O of a the points x with {a(v )i{x}} is-an-element-of O are dense in the subspace topology. Then A i s a polyhedron whose vertices are rational points.