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|≪n2, one can always find two co-Sidon subsets A′,B′ with sizes |A′|=k,|B′|=ℓ, whenever k,ℓ satisfy kℓ2≪n4/E. An example showing that this is best possible up to the logarithmic factor is presented. When the energy is large, E≫n3, we show that there exist co-Sidon subsets A′,B′ of A,B with sizes |A′|=k,|B′|=ℓ whenever k,ℓ satisfy kℓ≪n and show that this is best possible. These results are extended (non-optimally, however) to the full range of values of E.