On a problem of partitions of the set of nonnegative integers with the same representation functions

Dombi has shown that the set $\mathbb{N}$ of all non-negative integers can be partitioned into two subsets with identical representation functions. In this paper, we prove that one cannot partition $\mathbb{N}$ into more than two subsets with identical representation functions, while for any integer $k\ge 3$ there is a partition $\mathbb{N}=A_1\cup ?\cup A_k$ such that $A_i$ and $A_{k+1?i}$ have the same representation function for any integer $1\le i\le k$.