On the generalisation of Roth's theorem

TL;DR

This paper explores generalizations of Roth's approximation theorem on proper adelic curves, demonstrating how stronger assumptions lead to improved inequalities and unifying results across different fields.

Contribution

It introduces new generalizations of Roth's theorem on adelic curves, connecting and extending previous results by Corvaja and Vojta under specific conditions.

Findings

Stronger inequalities achieved with tighter assumptions

Unified framework for fields with product formula and arithmetic function fields

Recovery of Corvaja's and Vojta's results as special cases

Abstract

We present two possible generalisations of Roth's approximation theorem on proper adelic curves, assuming some technical conditions on the behavior of the logarithmic absolute values. We illustrate how tightening such assumptions makes our inequalities stronger. As special cases we recover Corvaja's results [Cor97] for fields admitting a product formula, and Vojta's ones [Voj21] for arithmetic function fields.