EXPANDING POLYNOMIALS ON SETS WITH FEW PRODUCTS

In this note, we prove that if A is a finite set of real numbers such that |AA|=K|A|, then for every polynomial f∈R[x,y] we have that |f(A,A)|=Ω_(K,degf)(|A|²), unless f is of the form f(x,y)=g(M(x,y)) for some monomial M and some univariate polynomial g. This is sharp up to the dependence on K and the degree of f.