Partitions of finite nonnegative integer sets with identical representation functions

TL;DR

This paper characterizes the structure of two subsets of nonnegative integers that partition a finite interval with exactly two elements in common, sharing identical representation functions for all nonnegative integers.

Contribution

It provides a complete description of the structure of such partitions with identical representation functions and a specific intersection size.

Findings

Identifies the structure of subsets with equal representation functions

Characterizes partitions of finite intervals with two-element intersection

Establishes conditions for identical representation functions across subsets

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 $n=s+s'$ with $s, s'\in S$ and $s<s'$. 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 $n$.