## Long arithmetic progressions in sumsets: Thresholds and bounds

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- 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
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## 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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*Proceedings of London Mathematics Society*90 (2005), 273-296. szemvu2 E. Szemerédi and V. H. Vu, Finite and infinite arithmetic progressions in sumsets,

*to appear in Annals of Mathematics*.

*submitted*. Vunew2 V. H. Vu, New results concerning subset sums,

*in preparation*.

## 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