Note on a result of Chen and Lev (Q2322540)

Let \(\mathbb{N}\) be the set of all nonnegative integers. For a given set \(S\subset\mathbb{N}\) the representation function \(R_S(n)\) is defined as the number of solutions of the equation \(n=s+s'\), \(s<s'\), \(s,s'\in S\). Let \(l\) be a positive integer. For a finite or infinite set of positive integers \(\{h_1,h_2,\dots\}\) with \(h_1<h_2<\cdots\), the Hilbert cube is defined by \(\left\{\sum_i\varepsilon_i h_i\mid \varepsilon\in\{0,1\}\right\}\), where in each sum there are only finitely many \(\varepsilon_i\ne 0\).\par In this paper, using the notion of Hilbert cube, it is proved that there exist sets \(A\) and \(B\) such that \(\mathbb{N}=A\cup B\) and \(A\cap B=(2^{2 l}-1)+(2^{2 l+1}-1)\mathbb{N}\) and \(R_A(n)=R_B(n)\ge 1\) for every positive integer \(n\) except for finite exceptions. This result improves the result of \textit{Y.-G. Chen} and \textit{V. F. Lev} [Integers 16, Paper A36, 4 p. (2016; Zbl 1404.11013)].