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

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

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.**P. Erdos, Problems and results on combinatorial number theory III, in: M. B. Nathanson, editor,*Number Theory Day, New York 1976*,*Lecture Notes in Mathematics*, vol. 626, 1977, Springer-Verlag, Berlin, pp. 43-72. MR**57:12442****2.**P. Erdos, Problems and results in combinatorial analysis and combinatorial number theory, in: Y. Alavi, G. Chartrand, O. R. Ollerman, and A. J. Schwenk, editors,*Graph Theory, Combinatorics, and Applications*, 1991, John Wiley, New York, pp. 397-406. MR**93g:05136****3.**P. Erdos and E. Szemerédi, On sums and products of integers, in: P. Erdos, L. Alpár, G. Halász, and A. Sárközy, editors,*Studies in Pure Mathematics, To the Memory of Paul Turán*, 1983, Birkhäuser Verlag, Basel, pp. 213-218. MR**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

Email:
nathansn@alpha.lehman.cuny.edu

DOI:
https://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