Partitions of finite nonnegative integer sets with identical representation functions

Abstract

Let $\mathbb{N}$ be the set of all nonnegative integers. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let the representation function $R_S(n)$ denote the number of solutions of the equation $n=s+s^{\prime }$ with $s,s^{\prime }\in S$ and $s<s^{\prime }$. In this paper, we determine the structure of $C,D\subseteq \mathbb{N}$ with $C\cup D=[0,m]$ and $|C\cap D|=2$ such that $R_C(n)=R_D(n)$ for any nonnegative integer $n$.

arXiv admin note: text overlap with arXiv:2111.07754, arXiv:1606.06929 by other authors