A note on representation functions with different weights (Q2787120)

For any positive integer \(k\) and any set \(A\) of nonnegative integers, let \(r_{1,k}(A,n)\) denote the number of solutions \((a_1, a_2)\) of the equation \(n=a_1+ k a_2\) with \(a_1, a_2\in A\). Let \(k,\ell\geq 2\) be two distinct integers. The authors prove that there exists a set \(A\subseteq N\) such that both \(r_{1,k}(A,n)=r_{1,k}(\mathbb N\setminus A,n)\) and \(r_{1,\ell}(A, n)=r_{1,\ell}(\mathbb N\setminus A,n)\) hold for all \(n\geq n_0\) if and only if \(\log k/\log \ell=a/b\) for some odd positive integers \(a,b\), disproving a conjecture of \textit{Q.-H. Yang} [Colloq. Math. 137, No. 1, 1--6 (2014; Zbl 1346.11011)]. They also show that for any set \(A\subseteq \mathbb N\) satisfying \(r_{1,k}(A, n)=r_{1,k}(\mathbb N\setminus A, n)\) for all \(n\geq n_0\), then \(r_{1,k}(A, n)\rightarrow \infty\) as \(n\rightarrow \infty\).