Three-Variable Expanding Polynomials and Higher-Dimensional Distinct Distances

We determine which quadratic polynomials in three variables are expanders over an arbitrary field F. More precisely, we prove that for a quadratic polynomial f F[x,y,z], which is not of the form g(h(x)+k(y)+l(z)), we have |f(AxBxC)|>> N-3/2 for any sets A,B,C < subset of> 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+A(2)| and max{|A+A|, |A(2)+A(2)|}, and we prove that a Cartesian product Ax...xA < subset of> Fd determines almost |A|(2) distinct distances if |A| is not too large.