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

Full-text PDF Free Access

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]**G. Elekes,*On the number of sums and products*, Acta Arithmetica 81, Fase 4 (1997), 365-367. MR**98h:11026****[E-N-R]**G. Elekes, M. Nathanson, I. Rusza,*Convexity and sumsets*, J. Number Theory**83**(2000), 194-201. MR**2001e:11020****[E-S]**P. Erdos, E. Szemerédi,*On sums and products of integers*, in P. Erdös, L. Alpár, G. Halász (editors), Studies in Pure Mathematics; in memory of P. Turán, pp. 213-218.MR**86m:11011****[Go]**W. T. Gowers,*A new proof of Szemerédi's theorem for arithmetic progressions of length 4*, GAFA 8 (1998), 529-551. MR**2000d:11019****[Ki]**S. V. Kisliakov,*Banach Spaces and Classical Harmonic Analysis*, in Handbook for the geometry of Banach Spaces,Vol. 1, pp. 871-898, North-Holland, Amsterdam, 2001. MR**2003d:46021****[K]**S. Konjagin,*Private communication.***[Na]**M. B. Nathanson,*Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer, 1996.*MR**98f:11011****[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.

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (1991):
05A99

Retrieve articles in all journals with MSC (1991): 05A99

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