A note on some Diophantine inequalities over adelic curves

TL;DR

This paper establishes an upper bound on the number of large solutions to a specific system of Diophantine inequalities over adelic curves, extending Roth's inequality without relying on the Northcott property.

Contribution

It introduces a new upper bound for solutions of Diophantine inequalities over adelic curves, generalizing Roth's inequality without the Northcott property assumption.

Findings

Upper bound on solutions without Northcott property

Extension of Roth's inequality to adelic curves

Applicable to systems of Diophantine inequalities

Abstract

Without assuming the Northcott property we provide an upper bound on the number of "big solutions" of a special system of Diophantine inequalities over proper adelic curves. This system is interesting since it is a stronger version Roth's inequality for adelic curves.