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

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- by Jean Bourgain and Mei-Chu Chang
- J. Amer. Math. Soc.
**17**(2004), 473-497 - DOI: https://doi.org/10.1090/S0894-0347-03-00446-6
- Published electronically: November 25, 2003
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## Abstract:

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 $N^{\epsilon }$.## References

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## Bibliographic Information

**Jean Bourgain**- Affiliation: Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540
- MR Author ID: 40280
- Email: bourgain@math.ias.edu
**Mei-Chu Chang**- Affiliation: Mathematics Department, University of California, Riverside, California 92521
- Email: mcc@math.ucr.edu
- Received by editor(s): September 5, 2003
- Published electronically: November 25, 2003
- © Copyright 2003 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**17**(2004), 473-497 - MSC (1991): Primary 05A99
- DOI: https://doi.org/10.1090/S0894-0347-03-00446-6
- MathSciNet review: 2051619