caa a22 4500 445358491 CHVBK 20180317142908.0 cr unu---uuuuu 170323e20111201xx s 000 0 eng 10.1007/s00493-011-2682-8 doi (NATIONALLICENCE)springer-10.1007/s00493-011-2682-8 On a problem of Cilleruelo and Nathanson [Elektronische Daten] [Yong-Gao Chen, Jin-Hui Fang] 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. János Bolyai Mathematical Society and Springer Verlag, 2011 Chen Yong-Gao School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, 210046, Nanjing, P. R. China aut Fang Jin-Hui Department of Mathematics, Nanjing University of Information Science & Technology, 210044, Nanjing, P. R. China aut Combinatorica Springer-Verlag 31/6(2011-12-01), 691-696 0209-9683 31:6<691 2011 31 493 https://doi.org/10.1007/s00493-011-2682-8 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00493-011-2682-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Chen Yong-Gao School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, 210046, Nanjing, P. R. China aut NATIONALLICENCE 700 1- Fang Jin-Hui Department of Mathematics, Nanjing University of Information Science & Technology, 210044, Nanjing, P. R. China aut NATIONALLICENCE 773 0- Combinatorica Springer-Verlag 31/6(2011-12-01), 691-696 0209-9683 31:6<691 2011 31 493 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer