An Elekes-Rónyai theorem for sets with few products

Given $d,n\in \mathbb{N}$, we write a polynomial $F\in \mathbb{C}[x_1,\dots ,x_n]$ to be degenerate if there exist $P\in \mathbb{C}[y_1,\dots ,y_{n-1}]$ and $m_j=x_1^{v_{j,1}}\dots x_n^{v_{j,n}}$ with $v_{j,1},\dots ,v_{j,n}\in \mathbb{Q}$, for every $1\le j\le n-1$, such that $F=P(m_1,\dots ,m_{n-1})$. Our main result shows that whenever $F$ is non-degenerate, then for every finite set $A\subseteq \mathbb{C}$ such that $\mathrm{\mid }A\cdot A\mathrm{\mid }\le K\mathrm{\mid }A\mathrm{\mid }$, one has $$\mathrm{\mid }F(A,\dots ,A)\mathrm{\mid }{\gg }_{d,n}\mathrm{\mid }A{\mathrm{\mid }}^n{2}^{-O_{d,n}((\log 2K{)}^{3+o(1)})}\mathrm{.}$$ This is sharp up to a factor of $O_{d,n,K}(1)$ since we have the upper bound $\mathrm{\mid }F(A,\dots ,A)\mathrm{\mid }\le \mathrm{\mid }A{\mathrm{\mid }}^n$ and the fact that for every degenerate $F$ and finite set $A\subseteq \mathbb{C}$ with $\mathrm{\mid }A\cdot A\mathrm{\mid }\le K\mathrm{\mid }A\mathrm{\mid }$, one has $$\mathrm{\mid }F(A,\dots ,A)\mathrm{\mid }\ll K^{O_F(1)}\mathrm{\mid }A{\mathrm{\mid }}^{n-1}\mathrm{.}$$ Our techniques rely on a variety of combinatorial and linear algebraic arguments combined with Freiman type inverse theorems and Schmidt’s subspace theorem.