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

Abstract

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. © 2013 Elsevier Ltd.