On sumsets of dissociated sets∗

Abstract

In the paper, we are studying some properties of subsets $Q\subseteq {\mathrm{\Lambda }}_1+\cdots +{\mathrm{\Lambda }}_k$, where ${\mathrm{\Lambda }}_i$ are dissociated sets. The exact upper bound for the number of solutions of the following equation:
$$\begin{array}{}\text{(1)}& q_1+\cdots +q_p=q_{p+1}+\cdots +q_{2p},q_i\in Q\end{array}$$
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.