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

For two finite sets of integers A and B their additive energy E ( A , B ) is the number of solutions to a + b = a ′ + b ′ , where a , a ′ ∈ A and b , b ′ ∈ B . Given finite sets A , B ⊆ Z with additive energy E ( A , B ) = | A | | B | + E , we investigate the sizes of largest subsets A ′ ⊆ A and B ′ ⊆ B with all | A ′ | | B ′ | sums a + b , a ∈ A ′ , b ∈ B ′ , being different (we call such subsets A ′ , B ′ co-Sidon). In particular, for | A | = | B | = n we show that in the case of small energy, n ⩽ E = E ( A , B ) − | A | | B | ≪ n 2 , one can always find two co-Sidon subsets A ′ , B ′ with sizes | A ′ | = k , | B ′ | = ℓ , whenever k , ℓ satisfy k ℓ 2 ≪ n 4 / E . An example showing that this is best possible up to the logarithmic factor is presented. When the energy is large, E ≫ n 3 , we show that there exist co-Sidon subsets A ′ , B ′ of A , B with sizes | A ′ | = k , | B ′ | = ℓ whenever k , ℓ satisfy k ℓ ≪ n and show that this is best possible. These results are extended (non-optimally, however) to the full range of values of E .