  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.
