An analogue of Liouville's Theorem and an application to cubic surfaces

TL;DR

This paper establishes a strong analogue of Liouville's Theorem for algebraic varieties and applies it to prove a conjecture related to cubic surfaces in projective three-space.

Contribution

It introduces a new Diophantine approximation theorem for algebraic varieties and confirms a conjecture for cubic surfaces.

Findings

Proved a strong Liouville-type theorem for algebraic varieties.

Validated a conjecture for cubic surfaces in projective space.

Extended Diophantine approximation techniques to new geometric contexts.

Abstract

We prove a strong analogue of Liouville's Theorem in Diophantine approximation for points on arbitrary algebraic varieties. We use this theorem to prove a conjecture of the first author for cubic surfaces in $\P^3$.