On the size of k-fold sum and product sets of integers

In this paper, we show that for all b > 1 there is a positive integer k=k (b) such that if A is an arbitrary finite set of integers, |A|=N>2, then either |kA|>N^b or |A^ (k) |>N^b. Here kA (resp. A^ (k) ) denotes the k-fold sum (resp. product) of A. This fact is deduced from the following harmonic analysis result obtained in the paper. For all q>2 and >0, there is a >0 such that if A satisfies |A A|< N^ |A|, then the q-constant ₐ (A) of A (in the sense of W. Rudin) is at most N^.