On sum-sets and product-sets of complex numbers
Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 921-924.

We give a simple argument that for any finite set of complex numbers A, the size of the the sum-set, A+A, or the product-set, A·A, is always large.

On donne une preuve simple que pour tout ensemble fini de nombres complexes A, la taille de l’ensemble de sommes A+A ou celle de l’ensemble de produits A·A est toujours grande.

Published online:
DOI: 10.5802/jtnb.527
József Solymosi 1

1 Department of Mathematics, University of British Columbia 1984 Mathematics Road, Vancouver, Colombie-Britannique, Canada V6T 1Z2
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József Solymosi. On sum-sets and product-sets of complex numbers. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 921-924. doi : 10.5802/jtnb.527. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.527/

[1] J. Bourgain,S. Konjagin, Estimates for the number of sums and products and for exponential sums over subgrups in finite fields of prime order. C. R. Acad. Sci. Paris 337 (2003), no. 2, 75–80. | MR: 1998834 | Zbl: 1041.11056

[2] J. Bourgain, N. Katz, T. Tao, A sum-product estimate in finite fields, and applications. Geometric And Functional Analysis GAFA 14 (2004), no. 1, 27–57. | MR: 2053599 | Zbl: 02121750

[3] M. Chang, A sum-product estimate in algebraic division algebras over R. Israel Journal of Mathematics (to appear).

[4] M. Chang, Factorization in generalized arithmetic progressions and applications to the Erdős-Szemerédi sum-product problems. Geometric And Functional Analysis GAFA 13 (2003), no. 4, 720–736. | MR: 2006555 | Zbl: 1029.11006

[5] M. Chang, Erdős-Szemerédi sum-product problem. Annals of Math. 157 (2003), 939–957. | MR: 1983786 | Zbl: 1055.11017

[6] Gy. Elekes, On the number of sums and products. Acta Arithmetica 81 (1997), 365–367. | MR: 1472816 | Zbl: 0887.11012

[7] P. Erdős, E. Szemerédi, On sums and products of integers. In: Studies in Pure Mathematics; To the memory of Paul Turán. P.Erdős, L.Alpár, and G.Halász, editors. Akadémiai Kiadó – Birkhauser Verlag, Budapest – Basel-Boston, Mass. 1983, 213–218. | MR: 820223 | Zbl: 0526.10011

[8] K. Ford, Sums and products from a finite set of real numbers. Ramanujan Journal, 2 (1998), (1-2), 59–66. | MR: 1642873 | Zbl: 0908.11008

[9] M. B. Nathanson, On sums and products of integers. Proc. Am. Math. Soc. 125 (1997), (1-2), 9–16. | MR: 1343715 | Zbl: 0869.11010

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