On the structure of partition which the difference of their representation function is a constant

Abstract

Let $\mathbb {N}$ be the set of nonnegative integers. For any set $A \subset \mathbb {N}$, let $R_1(A, n)$, $R_2(A, n)$ and $R_3(A, n)$ be the number of representations of n as $n=a+a',a, a'\in A$; $n=a+a',a, a'\in A$, $a<a'$; $n=a+a',a, a'\in A$, $a\le a'$ respectively. In this paper, for each $i\in \{2,3\}$, we determine the structure of the partition $\mathbb {N}=A\cup B$ with $A\cap B=\emptyset$ such that $R_i(A,n)-R_i(B,n)=k$ holds for all sufficiently large integers n, where k is a given integer.