Abstract
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.
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References
J. Cilleruelo, M. B. Nathanson: Perfect difference sets constructed from Sidon sets, Combinatorica 28(4) (2008), 401–414.
J. Cilleruelo and M. B. Nathanson: Dense sets of integers with prescribed representation functions, arXiv:0708.2853v1, 2007.
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I. Z. Ruzsa: An infinite Sidon sequence, J. Number Theory 68(1) (1998), 63–71.
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Supported by the National Natural Science Foundation of China, Grant No. 11071121 and Natural Science Foundation of the Jiangsu Higher Education Institutions, Grant No. 11KJB110006.