Let ℤ denote the set of all integers and ℕ the set of all positive integers. Let A be a set of integers. For every integer u, we denote by d A (u) and s A (u) the number of solutions of u=a − a′ with a,a′ ∈ A and u=a+a′ with a,a′ ∈ A and a≤a′, respectively.
Recently, J. Cilleruelo and M. B. Nathanson in [Perfect difference sets constructed from Sidon sets, Combinatorica 28 (4) (2008), 401–414] posed the following problem: Given two functions f 1: ℕ→ℕ and f 2: ℤ→ℕ. Is the condition lim inf u→∞ f 1(u)≥2 and lim inf |u|→∞ f 2(u)≥2 sufficient to assure that there exists a set A such that d A (n)=f 1(n) for all n∈ ℕ and s A (n)=f 2(n) for all n∈ ℔?
We prove that the answer to this problem is affirmative.