# Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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## Long arithmetic progressions in sumsets: Thresholds and boundsHTML articles powered by AMS MathViewer

by E. Szemerédi and V. Vu
J. Amer. Math. Soc. 19 (2006), 119-169
DOI: https://doi.org/10.1090/S0894-0347-05-00502-3
Published electronically: September 13, 2005

## Abstract:

For a set $A$ of integers, the sumset $lA =A+\dots +A$ consists of those numbers which can be represented as a sum of $l$ elements of $A$: $lA =\{a_1+\dots + a_l| a_i \in A_i \}.$ Closely related and equally interesting notion is that of $l^{\ast }A$, which is the collection of numbers which can be represented as a sum of $l$ different elements of $A$: $l^{\ast }A =\{a_1+\dots + a_l| a_i \in A_i, a_i \neq a_j \}.$ The goal of this paper is to investigate the structure of $lA$ and $l^{\ast }A$, where $A$ is a subset of $\{1,2, \dots , n\}$. As application, we solve two conjectures by Erdös and Folkman, posed in 1960s.
References
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Bibliographic Information
• E. Szemerédi
• Affiliation: Department of Computer Science, Rutgers University, New Brunswick, New Jersey 08854
• Email: szemered@cs.rutgers.edu
• V. Vu
• Affiliation: Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112
• Email: vanvu@ucsd.edu, vanvu@math.rutgers.edu
• Received by editor(s): October 6, 2004
• Published electronically: September 13, 2005
• Additional Notes: The first author is supported by an NSF grant.
The second author is an A. Sloan Fellow and is supported by an NSF Career Grant.
• © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
• Journal: J. Amer. Math. Soc. 19 (2006), 119-169
• MSC (2000): Primary 11B25, 11P70, 11B75
• DOI: https://doi.org/10.1090/S0894-0347-05-00502-3
• MathSciNet review: 2169044