Finding large co-Sidon subsets in sets with a given additive energy

On an application of higher energies to Sidon sets

“…First of all, notice that if S is a random subset of [N ], which was obtained piking elements from [N ] independently at random with probability q ∼ N −1/2 , then with probability 1 − o(1) one has r S−S (x) ≪ log N see, e.g., [5,Lemma 4.3] and a similar lower bound for the function r S−S (x) takes place. Thus for any fixed g sets belonging to B • 2 [g] are far from random sets.…”

“…. , a) ∈ A g : a ∈ A} ⊆ G g , then (18) means that the sum S g and ∆ g (S) is direct, in other words, S g and ∆ g (S) form a co-Sidon pair, see [5]. Further formula (18) suggests us the following definition for B…”