On the supremum of the representation function of a sumset

Abstract

Let A be a subset of the set of nonnegative integers ℕ ∪ {0}, and let r_A (n) be the number of representations of n ≥ 0 by the sum a + b with a, b ∈ A. Define s(A):= sup_n≥0 r _A (n) for each A ⊆ ℕ ∪ {0}. Given any three numbers a, b, c ∈ {2, 3, 4,…} ∪ {∞} satisfying max(a, b) ≤ c we prove that there exist two infinite disjoint subsets A, B of N such that for each n ∈ ℕ the nth element of A does not exceed the nth element of B, s(A) = a, s(B) = b and s(A ∪ B) = c. This generalizes two results of Grekos, Haddad, Helou and Pihko.