An Elekes–Rónyai Theorem for Sets With Few Products

Given n ∈ N, we call a polynomial F ∈ C[x₁, ,xₙ] degenerate if there exist P∈ C[y₁, , yₙ₋₁] and monomials m₁, , mₙ₋₁ with fractional exponents, such that F = P(m₁, , mₙ₋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^-Od,n((log 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^OF(1)|A|ⁿ⁻¹.align*Our techniques rely on Freiman type inverse theorems and Schmidt’s subspace theorem.