An Elekes–Rónyai Theorem for Sets With Few Products

Given $n \in \mathbb{N}$, we call a polynomial $F \in \mathbb{C}[x_{1},\dots ,x_{n}]$ degenerate if there exist $P\in \mathbb{C}[y_{1}, \dots , y_{n-1}]$ and monomials $m_{1}, \dots , m_{n-1}$ with fractional exponents, such that $F = P(m_{1}, \dots , m_{n-1})$. Our main result shows that whenever a polynomial $F$, with degree $d \geq 1$, is non-degenerate, then for every finite, non-empty set $A\subset \mathbb{C}$ such that $|A\cdot A| \leq K|A|$, one has $$ \begin{align*} & |F(A, \dots, A)| \gg |A|^{n} 2^{-O_{d,n}((\log 2K)^{3 + o(1)})}. \end{align*} $$This is sharp since for every degenerate $F$ and finite set $A \subset \mathbb{C}$ with $|A\cdot A| \leq K|A|$, one has $$ \begin{align*} & |F(A,\dots,A)| \ll K^{O_{F}(1)}|A|^{n-1}.\end{align*} $$Our techniques rely on Freiman type inverse theorems and Schmidt’s subspace theorem.