On sums and products of integers
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- by Melvyn B. Nathanson PDF
- Proc. Amer. Math. Soc. 125 (1997), 9-16 Request permission
Abstract:
Erdős 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 $A$. Erdős 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
- Paul Erdős, Problems and results on combinatorial number theory. III, Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) Lecture Notes in Math., Vol. 626, Springer, Berlin, 1977, pp. 43–72. MR 0472752
- 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
- P. Erdős and E. Szemerédi, On sums and products of integers, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 213–218. MR 820223
- M. B. Nathanson and G. Tenenbaum, Inverse theorems and the number of sums and products (to appear).
Additional Information
- Melvyn B. Nathanson
- Affiliation: Department of Mathematics, Lehman College (CUNY), Bronx, New York 10468
- Email: nathansn@alpha.lehman.cuny.edu
- 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
- © Copyright 1997 American Mathematical Society
- 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