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

Dombi has shown that the set 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 N into more than two subsets with identical representation functions, while for any integer k≥3 there is a partition N=A1∪⋯∪Ak such that Ai and Ak+1−i have the same representation function for any integer 1≤i≤k.