k-fold sums from a set with few products

In the present paper we show that if A is a set of n real numbers, and the product set A.A has at most n^(1+c) elements, then the k-fold sumset kA has at least n^(log(k/2)/2 log 2 + 1/2 - f_k(c)) elements, where f_k(c) -> 0 as c -> 0. We believe that the methods in this paper might lead to a much stronger result; indeed, using a result of Trevor Wooley on Vinogradov's Mean Value Theorem and the Tarry-Escott Problem, we show that if |A.A| < n^(1+c), then |k(A.A)| > n^(Omega((k/log k)^(1/3))), for c small enough in terms of k (we believe that a certain modification of this argument can perhaps produce similar conclusions for kA).