On a problem of Cilleruelo and Nathanson

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=aa′ with a,a′A and u=a+a′ with a,a′A and aa′, 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

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Correspondence to Yong-Gao Chen.

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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.

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Chen, YG., Fang, JH. On a problem of Cilleruelo and Nathanson. Combinatorica 31, 691–696 (2011). https://doi.org/10.1007/s00493-011-2682-8

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Mathematics Subject Classification (2000)

  • 11B13
  • 11B34