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On sums and products of integers


Author: Yong-Gao Chen
Journal: Proc. Amer. Math. Soc. 127 (1999), 1927-1933
MSC (1991): Primary 11B05, 11B13, 11B75, 11P99, 05A17
DOI: https://doi.org/10.1090/S0002-9939-99-04833-9
Published electronically: February 11, 1999
MathSciNet review: 1600124
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Abstract | References | Similar Articles | Additional Information

Abstract: Erdös and Szemerédi proved that if $A$ is a set of $k$ positive integers, then there must be at least $ck^{1+\delta}$ integers that can be written as the sum or product of two elements of $A$, where $c$ is a constant and $\delta>0$. Nathanson proved that the result holds for $\delta=\frac 1{31}$. In this paper it is proved that the result holds for $\delta=\frac 15$ and $c=\frac 1{20}$.


References [Enhancements On Off] (What's this?)

  • 1. P. Erdös, Problems and results on combinatorial number theory III, M. B. Nathanson editor, Number Theory Day, New York, 1976, Lecture Notes, vol. 626, Springer-Verlag, Berlin, pp. 43-72. MR 57:12442
  • 2. P. Erdös and E. Szemerédi, On sums and products of integers, P. Erdös, L. Alpár, G. Halász, and A. Sárközy, editors, Studies in Pure Mathematics, To the Memory of Paul Turán, Birkhäuser Verlag, Basel, pp. 213-218, 1983. MR 86m:11011
  • 3. M. B. Nathanson, On sums and products of integers, Proc. Amer. Math. Soc. 125 (1997), 9-15. MR 97c:11010
  • 4. M. B. Nathanson and G. Tenenbaum, Inverse theorems and the number of sums and products (to appear).

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Additional Information

Yong-Gao Chen
Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China
Email: ygchen@pine.njnu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-99-04833-9
Keywords: Additive number theory, sumsets, sums and products of integers
Received by editor(s): September 24, 1997
Published electronically: February 11, 1999
Additional Notes: This research was supported by the Fok Ying Tung Education Foundation and the National Natural Science Foundation of China
Communicated by: David E. Rohrlich
Article copyright: © Copyright 1999 American Mathematical Society

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