On sumsets of dissociated sets

In the paper we are studying some properties of subsets $Q subset Lambda_1+···+Lambda_k$, where $Lambda_i$ are dissociated sets. The exact upper bound for the number of solutions of the following equation $q_1 +···+q_p =q_{p+1} +dots+q_{2p}, q_i in Q~~~~~ (1)$ in groups $mathbb F_n^2$ is found. Using our approach, we easily prove a recent result of J. Bourgain on sets of large exponential sums and obtain a tiny improvement of his theorem. Besides an inverse problem is considered in the article. Let $Q$ be a set belonging to a sumset of two dissociated sets such that equation $(1)$ has many solutions. We prove that in the case the large proportion of $Q$ is highly structured.