Additive structures in sumsets
arXiv:math/0605520 · doi:10.1017/S030500410700093X
Abstract
Suppose that A is a subset of the integers {1,...,N} of density a. We provide a new proof of a result of Green which shows that A+A contains an arithmetic progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore we improve the length of progression guaranteed in higher sumsets; for example we show that A+A+A contains a progression of length roughly N^{ca} improving on the previous best of N^{ca^{2+ε}}.
28 pp. Corrected typos. Updated references.