ИСТИНА

A weak Sidon set S_ k of degree k>1 is a set having no solutions of the equation (*) x_1+...+x_k = x'_1+...+x'_k, where variables x_1, ..., x_k, x'_1, ..., x'_k \in S_k are different. Determining the maximal size of such a set from the segment {1,....N} is a rather old question of Additive Combinatorics having a little success. Recently, bounding the number of the solutions equation (*), Schoen and Shkredov showed that |S_k| \ll k^{2-c} N^{1/k}, where c>0 is an absolute constant. We give a scheme of the proof in our talk.