Additive structures in sumsets

الأصل · EN

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 Nca improving on the previous best of Nca²⁺ε.