A further improvement of the quantitative Subspace Theorem

TL;DR

This paper advances the quantitative bounds in the Absolute Subspace Theorem by refining previous results and combining methods from key proofs in Diophantine approximation, leading to tighter explicit bounds.

Contribution

It improves the quantitative version of the Absolute Parametric Subspace Theorem and derives a sharper quantitative version of the Absolute Subspace Theorem.

Findings

Enhanced explicit upper bounds for solutions of Diophantine inequalities.

Refined techniques combining Evertse-Schlickewei and Faltings-Wuestholz methods.

Improved understanding of the distribution of solutions in subspaces.

Abstract

In 2002, Evertse and Schlickewei 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, with ideas from Faltings' and Wuestholz' proof of the Subspace Theorem.