The inverse sieve problem for algebraic varieties over global fields

Abstract

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 $S\subseteq Z$ 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 the main result of Walsh in [Duke Math. J., vol.161, (2012), 2001-2022].

v3: More detailed proofs and explanations. Theorem 1.6 has been slightly modified thanks to a comment of the referee. Final version. To appear in Revista Matemática Iberoamericana