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

For two finite sets of integers AA and BB their additive energy E(A,B) is the number of solutions to a+b=a′+b′a+b=a′+b′, where a,a′∈Aa,a′∈A and b,b′∈Bb,b′∈B. Given finite sets A,B⊆ZA,B⊆Z with additive energy E(A,B)=|A||B|+E, we investigate the sizes of largest subsets A′⊆AA′⊆A and B′⊆BB′⊆B with all |A′||B′||A′||B′| sums a+ba+b, a∈A′,b∈B′a∈A′,b∈B′, being different (we call such subsets A′,B′A′,B′co-Sidon  ). In particular, for |A|=|B|=n|A|=|B|=n we show that in the case of small energy, n⩽E=E(A,B)−|A||B|≪n2, one can always find two co-Sidon subsets A′,B′A′,B′ with sizes |A′|=k,|B′|=ℓ|A′|=k,|B′|=ℓ, whenever k,ℓk,ℓ satisfy kℓ2≪n4/Ekℓ2≪n4/E. An example showing that this is best possible up to the logarithmic factor is presented. When the energy is large, E≫n3E≫n3, we show that there exist co-Sidon subsets A′,B′A′,B′ of A,BA,B with sizes |A′|=k,|B′|=ℓ|A′|=k,|B′|=ℓ whenever k,ℓk,ℓ satisfy kℓ≪nkℓ≪n and show that this is best possible. These results are extended (non-optimally, however) to the full range of values of EE.