Products of varieties with many integral points

Corvaja and Zannier asked whether a smooth projective integral variety with a dense set of rational points over a number field satisfies the weak Hilbert property. We introduce an extension of the weak Hilbert property for schemes over arithmetic base rings by considering near-integral points, extending Corvaja–Zannier’s question beyond the projective case. Building on work of Bary-Soroker–Fehm–Petersen and Corvaja–Demeio–Javanpeykar–Lombardo–Zannier, we prove several properties of this more general notion, in particular its persistence under products. We also answer positively Corvaja–Zannier’s question for all algebraic groups over finitely generated fields of characteristic zero.