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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On sums and products of integers

Author: Melvyn B. Nathanson
Journal: Proc. Amer. Math. Soc. 125 (1997), 9-16
MSC (1991): Primary 11B05, 11B13, 11B75, 11P99, 05A17
MathSciNet review: 1343715
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Abstract | References | Similar Articles | Additional Information

Abstract: Erdos and Szemerédi conjectured that if $A$ is a set of $k$ positive integers, then there must be at least $k^{2-\varepsilon }$ integers that can be written as the sum or product of two elements of $A$. Erdos and Szemerédi proved that this number must be at least $c k^{1 + \delta }$ for some $\delta > 0$ and $k \geq k_0$. In this paper it is proved that the result holds for $\delta = 1/31$.

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

  • 1. Paul Erdős, Problems and results on combinatorial number theory. III, Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), Springer, Berlin, 1977, pp. 43–72. Lecture Notes in Math., Vol. 626. MR 0472752 (57 #12442)
  • 2. P. Erdös, Problems and results in combinatorial analysis and combinatorial number theory, Graph theory, combinatorics, and applications, Vol.\ 1 (Kalamazoo, MI, 1988), Wiley-Intersci. Publ., Wiley, New York, 1991, pp. 397–406. MR 1170793 (93g:05136)
  • 3. P. Erdős and E. Szemerédi, On sums and products of integers, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 213–218. MR 820223 (86m:11011)
  • 4. M. B. Nathanson and G. Tenenbaum, Inverse theorems and the number of sums and products (to appear).

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Additional Information

Melvyn B. Nathanson
Affiliation: Department of Mathematics, Lehman College (CUNY), Bronx, New York 10468

PII: S 0002-9939(97)03510-7
Keywords: Additive number theory, sumsets, sums and products of integers
Received by editor(s): June 25, 1994
Received by editor(s) in revised form: May 23, 1995
Additional Notes: This work was supported in part by grants from the PSC-CUNY Research Award Program and the National Security Agency Mathematical Sciences Program
Communicated by: William W. Adams
Article copyright: © Copyright 1997 American Mathematical Society

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