Multiplicative complements. II. (Q6613256)

The paper under review is a sequel to the authors' recent work [\textit{A. Kocsis} et al., Acta Arith. 207, No. 2, 101--120 (2023; Zbl 1522.11094)] where they study multiplicative complement of subsets of the set of positive integers. Letting \(\mathbb{Z}^+\) to be the set of positive integers and \(A_i \subseteq \mathbb{Z}^+\) for every \(1\le i\le h\), the \(h\)-tuple \((A_1,\dots ,A_h)\) form multiplicative complements of order \(h\) if every positive integers \(n\) can be written as \(n=a_1\cdots a_h\), \(a_i\in A_i\). The set of multiplicative complements of order \(h\) is denoted by \(MC_h\). In the present paper, the authors prove several results, mutually estimating the quantities\N\[\Nm_{A,B}(x)=\min \{ A(x),B(x) \},\quad M_{A,B}(x)=\max \{ A(x),B(x) \},\N\]\Nfor pairs of sets \(A,B\in MC_2\), where \(A(x)=\sum_{n\in A}1\) is the counting function of \(A\). They prove that for every infinite \((A,B)\in MC_2\),\N\[\N\liminf_{x\to \infty}\frac{M_{A,B}(x)\log m_{A,B}(x)}{x}>0.5,\quad \limsup_{x\to \infty}\frac{M_{A,B}(x)\log m_{A,B}(x)}{x}=\infty.\N\]\NLetting \(f(x)\) to be a function such that \(f(x)\to \infty \) as \(x\to \infty \), they show that there exists infinite \((A,B)\in MC_2\) such that \(A(x)=o(x)\), \(B(x)=o(x)\) and\N\[\N\liminf_{x\to \infty }\frac{m_{A,B}(x)}{f(x)}=0.\N\]\NThe authors prove that if we assume that for some function \(f(x)>0\), \(x\ge 1\), the series \(\sum_{n=1}^{\infty} f(n)/n^2\) converges, then there is no \((A,B)\in MC_2\) such that \(A(x)=O(f(x))\) and \(B(x)=o(x)\). Also, they show that there exists \((A,B)\in MC_2\) such that \(M_{A,B}(x)\sqrt{\log m_{A,B}(x)}\sim x/\sqrt{\pi}\) as \(x\to \infty\), and end the paper by proposing some related open problems.