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: Erdos and Szemerédi conjectured that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of . Erdos and Szemerédi proved that this number must be at least for some and . In this paper it is proved that the result holds for .

**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****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****3.**P. Erdős and E. Szemerédi,*On sums and products of integers*, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 213–218. MR**820223****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

Email:
nathansn@alpha.lehman.cuny.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-97-03510-7

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