On sums and products of integers
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- by Yong-Gao Chen PDF
- Proc. Amer. Math. Soc. 127 (1999), 1927-1933 Request permission
Abstract:
Erdös and Szemerédi proved that if $A$ is a set of $k$ positive integers, then there must be at least $ck^{1+\delta }$ integers that can be written as the sum or product of two elements of $A$, where $c$ is a constant and $\delta >0$. Nathanson proved that the result holds for $\delta =\frac 1{31}$. In this paper it is proved that the result holds for $\delta =\frac 15$ and $c=\frac 1{20}$.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. Erdős and E. Szemerédi, On sums and products of integers, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 213–218. MR 820223
- Melvyn B. Nathanson, On sums and products of integers, Proc. Amer. Math. Soc. 125 (1997), no. 1, 9–16. MR 1343715, DOI 10.1090/S0002-9939-97-03510-7
- M. B. Nathanson and G. Tenenbaum, Inverse theorems and the number of sums and products (to appear).
Additional Information
- Yong-Gao Chen
- Affiliation: Department of Mathematics, Nanjing Normal University, Nanjing 210097, People’s Republic of China
- MR Author ID: 304097
- Email: ygchen@pine.njnu.edu.cn
- Received by editor(s): September 24, 1997
- Published electronically: February 11, 1999
- Additional Notes: This research was supported by the Fok Ying Tung Education Foundation and the National Natural Science Foundation of China
- Communicated by: David E. Rohrlich
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1927-1933
- MSC (1991): Primary 11B05, 11B13, 11B75, 11P99, 05A17
- DOI: https://doi.org/10.1090/S0002-9939-99-04833-9
- MathSciNet review: 1600124