On the size of $k$fold sum and product sets of integers
Authors:
Jean Bourgain and MeiChu Chang
Journal:
J. Amer. Math. Soc. 17 (2004), 473497
MSC (1991):
Primary 05A99
DOI:
https://doi.org/10.1090/S0894034703004466
Published electronically:
November 25, 2003
MathSciNet review:
2051619
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
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 $N^{\epsilon }$.

[BK]BK 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]Ch M. Chang, Erdős Szemerédi sumproduct problem, Annals of Math. 157 (2003), 939957.
 György Elekes, On the number of sums and products, Acta Arith. 81 (1997), no. 4, 365–367. MR 1472816, DOI https://doi.org/10.4064/aa814365367
 György Elekes, Melvyn B. Nathanson, and Imre Z. Ruzsa, Convexity and sumsets, J. Number Theory 83 (2000), no. 2, 194–201. MR 1772612, DOI https://doi.org/10.1006/jnth.1999.2386
 P. Erdős and E. Szemerédi, On sums and products of integers, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 213–218. MR 820223
 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, DOI https://doi.org/10.1007/s000390050065
 S. V. Kislyakov, Banach spaces and classical harmonic analysis, Handbook of the geometry of Banach spaces, Vol. I, NorthHolland, Amsterdam, 2001, pp. 871–898. MR 1863708, DOI https://doi.org/10.1016/S18745849%2801%2980022X [K]K S. Konjagin, Private communication.
 Melvyn B. Nathanson, Additive number theory, Graduate Texts in Mathematics, vol. 165, SpringerVerlag, New York, 1996. Inverse problems and the geometry of sumsets. MR 1477155 [Ru]Ru W. Rudin, Trigonometric series with gaps, J. Math. Mech. 9 (1960), 203–227. [So]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
MR Author ID:
40280
Email:
bourgain@math.ias.edu
MeiChu 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
Article copyright:
© Copyright 2003
American Mathematical Society