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

Bourgain, Jean ¹   ; Chang, Mei-Chu ²

¹ Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540
² Mathematics Department, University of California, Riverside, California 92521

Journal of the American Mathematical Society, Tome 17 (2004) no. 2, pp. 473-497

Résumé

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 $\epsilon >0$, there is a $\delta >0$ such that if $A$ satisfies $|A \cdot A| N^{\delta }|A|$, then the $\lambda _q$-constant $\lambda _{q}(A)$ of $A$ (in the sense of W. Rudin) is at most $N^{\epsilon }$.

Détail

DOI : 10.1090/S0894-0347-03-00446-6

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