The inverse sieve problem for algebraic varieties over global fields

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 subset of Z of rational points of bounded height occupies few residue classes modulo p for many prime ideals 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 S subset of {0, ..., N}(d).