Long arithmetic progressions in sumsets: Thresholds and bounds

Szemerédi, E.; Vu, V.

doi:10.1090/S0894-0347-05-00502-3

Long arithmetic progressions in sumsets: Thresholds and bounds

Authors: E. Szemerédi and V. Vu
Journal: J. Amer. Math. Soc. 19 (2006), 119-169
MSC (2000): Primary 11B25, 11P70, 11B75
Published electronically: September 13, 2005
MathSciNet review: 2169044
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Abstract: For a set of integers, the sumset $lA =A+\dots+A$ consists of those numbers which can be represented as a sum of elements of :

$\begin{displaymath}lA =\{a_1+\dots + a_l\vert a_i \in A_i \}. \end{displaymath}$

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 different elements of :

$\begin{displaymath}l^{\ast}A =\{a_1+\dots + a_l\vert a_i \in A_i, a_i \neq a_j \}. \end{displaymath}$

The goal of this paper is to investigate the structure of and $l^{\ast}A$ , where is a subset of $\{1,2, \dots, n\}$ . As application, we solve two conjectures by Erdös and Folkman, posed in 1960s.

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