Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances

Abstract

We determine which quadratic polynomials in three variables are expanders over an arbitrary field $\mathbb{F}$. More precisely, we prove that for a quadratic polynomial f ∈ $\mathbb{F}$[x,y,z], which is not of the form g(h(x)+k(y)+l(z)), we have |f(A×B×C)|≫N^3/2 for any sets A,B,C ⊂ $\mathbb{F}$ with |A|=|B|=|C|=N, with N not too large compared to the characteristic of F.

We give several related proofs involving similar ideas. We obtain new lower bounds on |A+A²| and max{|A+A|, |A²+A²|}, and we prove that a Cartesian product A×...×A ⊂ $\mathbb{F}^d$ determines almost |A|² distinct distances if |A| is not too large.