Partitions of $\mathbf{Z}_m$ with the Same Weighted Representation Functions

Abstract

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 $A\subset\mathbf{Z}_m$ and any $n\in\mathbf{Z}_m$, 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 $A\subset \mathbf{Z}_m$ satisifying $r_{A}^{\mathbf{k}}=r_{\mathbf{Z}_m\backslash A}^{\mathbf{k}}$. This settles a problem of Yang and Chen.