Partition problems in additive number theory (Q1907856)

Let \(A\) be a subset of integers. Let \[ \sum(A)= \Biggl\{ \sum_{b\in B} b: B\text{ is a non-empty finite subset of }A\Biggr\}. \] Let \(f_k(n)\) be the minimal integer such that if \([f_k(n)]= \bigcup^k_{i=1} A_i\) then \(n\in \bigcup^k_{i= 1}\sum (A_i)\). In a previous paper [\textit{B. Bollobás}, \textit{P. Erdős} and \textit{G. Jin}, Acta Arith. 64, 341-355 (1993; Zbl 0789.11007)] it was proven that \(f_2(n)= 2\sqrt n+ o(\sqrt n)\). In the present paper, the authors investigate the function \(f_k(n)\) for \(k>2\). They prove \(f_3(n)= (2\sqrt 2+ o(1))\sqrt n\) if \(n\) is an odd number and \(f_3(n)= (\sqrt 6+ o(1))\sqrt n\) if \(n\) is an even number. Furthermore, they get a general result proving: Let \(k>1\) be fixed. Then \(f_k(n)= (\eta_k(n)+ o(1))\sqrt n\), where the function \(\eta_k(n)\) depends only on the arithmetical structure of \(n\). Related questions are also investigated.