A further improvement of the Quantitative Subspace Theorem

Abstract

In 2002, Evertse and Schlickewei [11] obtained a quantitative version of the so-called Absolute Parametric Subspace Theorem. This result deals with a parametrized class of twisted heights. One of the consequences of this result is a quantitative version of the Absolute Subspace Theorem, giving an explicit upper bound for the number of subspaces containing the solutions of the Diophantine inequality under consideration. In the present paper, we further improve Evertse’s and Schlickewei’s quantitative version of the Absolute Parametric Subspace Theorem, and deduce an improved quantitative version of the Absolute Subspace Theorem. We combine ideas from the proof of Evertse and Schlickewei (which is basically a substantial refinement of Schmidt’s proof of his Subspace Theorem from 1972 [22]), with ideas from Faltings’ and Wustholz’ proof of the Subspace Theorem [14]. A new feature is an “interval result,” which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the Subspace Theorem.