EXPANDING POLYNOMIALS ON SETS WITH FEW PRODUCTS. Issue 1 (26th November 2019) - Pohoata, Cosmin
Abstract: In this note, we prove that if A is a finite set of real numbers such that | A A | = K | A |, then for every polynomial f ∈ R [ x, y ], we have that | f ( A, A ) | = Ω K, deg f ( | 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 .