The Hilbert property for arithmetic schemes

TL;DR

This paper extends the Hilbert property to arithmetic schemes over integral domains, establishing that the set of near-integral points is non-thin and generalizing key results about their structure and covers.

Contribution

It introduces a new formulation of the Hilbert property for arithmetic schemes and generalizes existing results on their structure and covers.

Findings

The set of near-integral points is non-thin for schemes with the Hilbert property.

Generalization of structure results for products and finite étale covers.

Extension of the Hilbert property to a broader class of arithmetic schemes.

Abstract

We extend the usual Hilbert property for varieties over fields to arithmetic schemes over integral domains by demanding the set of near-integral points (as defined by Vojta) to be non-thin. We then generalize results of Bary-Soroker-Fehm-Petersen and Corvaja-Zannier by proving several structure results related to products and finite \'{e}tale covers of arithmetic schemes with the Hilbert property.