A generalization of Frieman's 3k-3 theorem

Abstract

We prove a generalization of Frieman's $3k-3$ theorem for the sumset $$ Σ^{l}(A_1,\ldots,A_k)=\{a_{j_{1}}+\cdots+a_{j_{l}}:\,1\leq j_{1}<\cdots<j_{l}\leq k,\ a_{j_{s}}\in A_{j_{s}}\text{ for all }s\}. $$

Seemingly the main result of this paper is not strong in most cases