Geometric Invariant Theory and Roth's Theorem

TL;DR

This paper offers a novel proof of Roth's Theorem and its variants by leveraging Geometric Invariant Theory, connecting height comparisons of semi-stable points with classical zero estimates.

Contribution

It introduces a GIT-based approach to prove Roth's Theorem, providing a geometric perspective that unifies various versions of the theorem.

Findings

Proof of Roth's Theorem via GIT height comparison

Extension to variants like Lang's and Vojta's moving targets

Geometric semi-stability replaces classical zero estimates

Abstract

We present a proof of Thue-Siegel-Roth's Theorem (and its more recent variants, such as those of Lang for number fields and that "with moving targets" of Vojta) as an application of Geometric Invariant Theory (GIT). Roth's Theorem is deduced from a general formula comparing the height of a semi-stable point and the height of its projection on the GIT quotient. In this setting, the role of the zero estimates appearing in the classical proof is played by the geometric semi-stability of the point to which we apply the formula.