Partitions of nonnegative integers with identical representation functions

TL;DR

This paper characterizes the conditions under which two partitions of nonnegative integers with identical representation functions exist, based on specific arithmetic progressions related to powers of two.

Contribution

It provides a complete characterization of partitions with identical representation functions involving particular arithmetic progressions and powers of two.

Findings

Existence of sets C and D with specified union and intersection properties.

Equality of representation functions for all n under certain conditions.

Explicit formulas for r1, r2, and m involving powers of two.

Abstract

Let $\mathbb{N}$ be the set of all nonnegative integers. For any integer $r$ and $m$, let $r+m\mathbb{N}=\{r+mk: k\in\mathbb{N}\}$. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s<s'$. Let $r_{1}, r_{2}, m$ be integers with $0<r_{1}<r_{2}<m$ and $2\mid r_{1}$. In this paper, we prove that there exist two sets $C$ and $D$ with $C\cup D=\mathbb{N}$ and $C\cap D=(r_{1}+m\mathbb{N})\cup (r_{2}+m\mathbb{N})$ such that $R_{C}(n)=R_{D}(n)$ for all $n\in\mathbb{N}$ if and only if there exists a positive integer $l$ such that $r_{1}=2^{2l+1}-2, r_{2}=2^{2l+1}-1, m=2^{2l+2}-2$.