CONICET | Buscador de Institutos y Recursos Humanos

Let $K$ be a global field and let $Z$ be a geometrically irreducible algebraic variety defined over $K$. We show that if a big set $Ssubseteq Z$ of rational points of bounded height occupies few residue classes modulo $mathfrak{p}$ for many prime ideals $mathfrak{p}$, then a positive proportion of $S$ must lie in the zero set of a polynomial of low degree that does not vanish at $Z$. This generalizes a result of Walsh who studied the case when $Ssubseteq {0,ldots ,N}^{d}$.