On the size of -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 | References | Similar Articles | Additional Information

Abstract: In this paper, we show that for all there is a positive integer such that if is an arbitrary finite set of integers, , then either or . Here (resp. ) denotes the -fold sum (resp. product) of . This fact is deduced from the following harmonic analysis result obtained in the paper. For all and , there is a such that if satisfies , then the -constant of (in the sense of W. Rudin) is at most .

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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