An Elekes–Rónyai Theorem for Sets With Few Products

Abstract Given n N, we call a polynomial F Cx₁, , x₍ degenerate if there exist P Cy₁, , y₍-₁ and monomials m₁, , m₍-₁ with fractional exponents, such that F = P (m₁, , m₍-₁). Our main result shows that whenever a polynomial F, with degree d 1, is non-degenerate, then for every finite, non-empty set A C such that |A A| K|A|, one has align* & |F (A, , A) | |A|^n 2^-O₃, ₍ ( (2K) ^{3 + o (1) ) }. align* This is sharp since for every degenerate F and finite set A C with |A A| K|A|, one has align* & |F (A, , A) | K^O₅ (1) |A|^n-1. align* Our techniques rely on Freiman type inverse theorems and Schmidt’s subspace theorem.