On sumsets of dissociated sets∗

In the paper, we are studying some properties of subsets \( Q \subseteq \Lambda_1 + \cdots + \Lambda_k \), where \( \Lambda_i \) are dissociated sets. The exact upper bound for the number of solutions of the following equation:
\[
q_1 + \cdots + q_p = q_{p+1} + \cdots + q_{2p}, \quad q_i \in Q \tag{1}
\]
in groups \( \mathbb{F}_2^n \) 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 subset of two dissociated sets such that equation (1) has many solutions. We prove that in this case, a large proportion of \( Q \) is highly structured.