An effective local-global principle and additive combinatorics in finite fields.

We use recent results about linking the number of zeros on algebraic varieties over ℂ, defined by polynomials with integer coefficients, and on their reductions modulo sufficiently large primes to study congruences with products and reciprocals of linear forms. This allows us to make some progress towards a question of B. Murphy, G. Petridis, O. Roche-Newton, M. Rudnev and I. D. Shkredov (2019) on an extreme case of the Erdős–Szemerédi conjecture in finite fields. [ABSTRACT FROM AUTHOR]