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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On the size of $k$-fold sum and product sets of integers
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by Jean Bourgain and Mei-Chu Chang PDF
J. Amer. Math. Soc. 17 (2004), 473-497 Request permission

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 $N^{\epsilon }$.
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Additional 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