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 .

**[B-K]**J. Bourgain, S. Konjagin,*Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order*, C. R. Acad. Sci. Paris Ser. I**337**(2003), 75-80.**[Ch]**M. Chang,*Erdos- Szemerédi sum-product problem*, Annals of Math.**157**(2003), 939-957.**[E]**György Elekes,*On the number of sums and products*, Acta Arith.**81**(1997), no. 4, 365–367. MR**1472816**, https://doi.org/10.4064/aa-81-4-365-367**[E-N-R]**György Elekes, Melvyn B. Nathanson, and Imre Z. Ruzsa,*Convexity and sumsets*, J. Number Theory**83**(2000), no. 2, 194–201. MR**1772612**, https://doi.org/10.1006/jnth.1999.2386**[E-S]**P. Erdős and E. Szemerédi,*On sums and products of integers*, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 213–218. MR**820223****[Go]**W. T. Gowers,*A new proof of Szemerédi’s theorem for arithmetic progressions of length four*, Geom. Funct. Anal.**8**(1998), no. 3, 529–551. MR**1631259**, https://doi.org/10.1007/s000390050065**[Ki]**S. V. Kislyakov,*Banach spaces and classical harmonic analysis*, Handbook of the geometry of Banach spaces, Vol. I, North-Holland, Amsterdam, 2001, pp. 871–898. MR**1863708**, https://doi.org/10.1016/S1874-5849(01)80022-X**[K]**S. Konjagin,*Private communication.***[Na]**Melvyn B. Nathanson,*Additive number theory*, Graduate Texts in Mathematics, vol. 165, Springer-Verlag, New York, 1996. Inverse problems and the geometry of sumsets. MR**1477155****[Ru]**W. Rudin,*Trigonometric series with gaps*, J. Math. Mech.**9**(1960), 203-227.**[So]**J. Solymosi,*On the number of sums and products,*preprint, 2003.

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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