# 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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J. Cilleruelo, M. B. Nathanson: Perfect difference sets constructed from Sidon sets, Combinatorica 28(4) (2008), 401–414.

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J. Cilleruelo and M. B. Nathanson: Dense sets of integers with prescribed representation functions, arXiv:0708.2853v1, 2007.

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M. B. Nathanson: The inverse problem for representation functions of additive bases, in: Number Theory: New York Seminar 2003, Springer, 2004, 253–262.

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M. B. Nathanson: Every function is the representation function of an additive basis for the integers, Port. Math. (N.S.) 62(1) (2005), 55–72.

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V. F. Lev: Reconstructing integer sets from their representation functions, Electron. J. Combin. 11(1) (2004), Research paper 78, 6pp (electronic).

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A. D. Pollington and C. Vanden: The integers as differences of a sequence, Canad. Bull. Math. 24(4) (1981), 497–499.

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I. Z. Ruzsa: An infinite Sidon sequence, J. Number Theory 68(1) (1998), 63–71.

## Author information

Authors

### Corresponding author

Correspondence to Yong-Gao Chen.

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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Reprints and Permissions

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