Partitions of \(\mathbb Z_m\) with the same weighted representation functions (Q405271)

Summary: Let \(\mathbf{k}=(k_1,k_2,\cdots,k_t)\) be a \(t\)-tuple of integers, and \(m\) be a positive integer. For a subset \(A\subset\mathbb Z_m\) and any \(n\in\mathbb Z_m\), let \(r_A^{\mathbf{k}}(n)\) denote the number of solutions of the equation \(k_1a_1+\cdots+k_ta_t=n\) with \(a_1,\cdots,a_t\in A\). In this paper, we give a necessary and sufficient condition on \((\mathbf{k},m)\) such that there exists a subset \(A\subset \mathbb Z_m\) satisifying \(r_{A}^{\mathbf{k}}=r_{\mathbb Z_m\backslash A}^{\mathbf{k}}\). This settles a problem of \textit{Q.-H. Yang} and \textit{Y.-G. Chen} [Taiwanese J. Math. 17, No. 4, 1311--1319 (2013; Zbl 1283.11024)].