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


Authors: Jean Bourgain and Mei-Chu Chang
Journal: J. Amer. Math. Soc. 17 (2004), 473-497
MSC (1991): Primary 05A99
DOI: https://doi.org/10.1090/S0894-0347-03-00446-6
Published electronically: November 25, 2003
MathSciNet review: 2051619
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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, $\vert A\vert=N>2$, 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 $N^{\epsilon }$.


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Additional Information

Jean Bourgain
Affiliation: Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540
Email: bourgain@math.ias.edu

Mei-Chu Chang
Affiliation: Mathematics Department, University of California, Riverside, California 92521
Email: mcc@math.ucr.edu

DOI: https://doi.org/10.1090/S0894-0347-03-00446-6
Received by editor(s): September 5, 2003
Published electronically: November 25, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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