On a Certain Generalization of the Balog-Szemerédi-Gowers Theorem

In the present paper we give a generalization of the Balog–Szemeredi–Gowers theorem for multiple sumsets A+A+⋯+A. Furthermore, the bounds we develop (on the rate of growth of certain subsums A′+A′+⋯+A′ as a function of k, the number of copies of A′ we add together) are almost the best possible.