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 6 E = E(A, B) − |A||B| ≪ n2, one can always find two coSidon 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 (nonoptimally, however) to the full range of values of E. © 2013 Elsevier Ltd. All rights reserved.