On qualitative aspects of the quantitative subspace theorem

TL;DR

This paper advances the understanding of the Diophantine exceptional set in the Subspace Theorem by introducing a new linear scattering result, utilizing linear sections and twisted height functions, with applications to number fields.

Contribution

It provides a novel linear scattering result for the Diophantine exceptional set, expanding and refining previous work, and develops new tools like linear sections and twisted height functions.

Findings

Established a new linear scattering type result for Diophantine exceptional sets.

Introduced the concept of logarithmic twisted height functions for local Weil functions.

Used Iitaka fibrations to analyze the asymptotic behavior of linear sections.

Abstract

We deduce Diophantine arithmetic inequalities for big linear systems and with respect to finite extensions of number fields. Our starting point is the Parametric Subspace Theorem, for linear forms, as formulated by Evertse and Ferretti \cite{Evertse:Ferretti:2013}. Among other features, this viewpoint allows for a partitioning of the linear scattering, for the Diophantine Exceptional set, that arises in the Subspace Theorem. Our perspective builds on our work \cite{Grieve:points:bounded:degree}, combined with earlier work of Evertse and Ferretti, \cite{Evertse:Ferretti:2013}, Evertse and Schlickewei, \cite{Evertse:Schlickewei:2002}, and others. As an application, we establish a novel linear scattering type result for the Diophantine exceptional set that arises in the main Diophantine arithmetic inequalities of Ru and Vojta \cite{Ru:Vojta:2016}. This result expands, refines and complements our earlier works (including \cite{Grieve:2018:autissier} and \cite{Grieve:points:bounded:degree}). A key tool to our approach is the concept of \emph{linear section} with respect to a linear system. This was defined in \cite{Grieve:points:bounded:degree}. Another point, which we develop in this article, is a notion of logarithmic \emph{twisted height functions} for local Weil functions and linear systems. As an additional observation, which is also of an independent interest, we use the theory of Iitaka fibrations to determine the asymptotic nature of such linear sections.