A New Diophantine Approximation Inequality on Surfaces and Its Applications

TL;DR

This paper establishes a new Diophantine approximation inequality on surfaces, generalizing recent results and applying it to problems involving integral points, gcd inequalities, and value distribution theory for holomorphic curves.

Contribution

It introduces a novel inequality for closed subschemes on surfaces, extending previous inequalities and connecting Diophantine approximation with complex analysis applications.

Findings

Proves a generalized Diophantine inequality for surfaces.

Applies the inequality to gcd problems and integral points on affine surfaces.

Derives a Second Main Theorem type inequality for holomorphic curves.

Abstract

We prove a Diophantine approximation inequality for closed subschemes on surfaces which can be viewed as a joint generalization of recent inequalities of Ru-Vojta and Heier-Levin in this context. As applications, we study various Diophantine problems on affine surfaces given as the complement of three numerically parallel ample projective curves: inequalities involving greatest common divisors, degeneracy of integral points, and related Diophantine equations including families of S-unit equations. We state analogous results in the complex analytic setting, where our main result is an inequality of Second Main Theorem type for surfaces, with applications to the study and value distribution theory of holomorphic curves in surfaces.