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, $\vert A\vert=N>2$, then either $\vert kA\vert>N^{b}$ or $\vert A^{(k)}\vert>N^{b}$. 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 $\epsilon >0$, there is a $\delta >0$ such that if $A$ satisfies $\vert A \cdot A\vert< N^{\delta }\vert A\vert$, then the $\lambda_q$-constant $\lambda _{q}(A)$ of $A$ (in the sense of W. Rudin) is at most $N^{\epsilon }$.