Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-97-03510-7
MathSciNet review: 1343715
Full-text PDF

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. 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).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11B05, 11B13, 11B75, 11P99, 05A17

Retrieve articles in all journals with MSC (1991): 11B05, 11B13, 11B75, 11P99, 05A17


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

American Mathematical Society