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|>Nᵇ or |A⁽ᵏ⁾|>Nᵇ. 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 ε >0, there is a δ >0 such that if A satisfies |A · A|< Nδ|A|, then the λ q-constant λ q(A) of A (in the sense of W. Rudin) is at most Nε.