Sum-product estimates via directed expanders

Abstract

Let Fq be a finite field of order q and P be a polynomial in Fq[x1,x2]. For a set A ⊂ Fq, define P(A) := {P(x1,x2)xi ∈ A}. Using certain constructions of expanders, we characterize all polynomials P for which the following holds If |A + A\ is small (compared to \A\), then \P(A)\ is large. The case P = x1x2 corresponds to the well-known sum-product problem. © International Press 2008.