# The inverse sieve problem for algebraic varieties over global fields

## TL;DR

This paper generalizes Walsh's inverse sieve result to algebraic varieties over global fields, showing that points with restricted residue class distribution are mostly contained in low-degree polynomial zero sets.

## Contribution

It extends inverse sieve methods to algebraic varieties over global fields, linking residue class distribution to algebraic structure.

## Key findings

- Points with limited residue class variation lie mostly in low-degree polynomial zero sets.
- The result applies to geometrically irreducible algebraic varieties over global fields.
- Generalizes Walsh's inverse sieve theorem to a broader algebraic setting.

## 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].