Partitions of Natural Numbers with the Intersection Not Empty

Let ℕ be the set of nonnegative integers. For a given set A ⊂ ℕ the representation functions R 2 ( A , n ) , R 3 ( A , n ) are defined as the number of solutions of the equation n = a + a ′ , a , a ′ ∈ A with condition a < a ′ , a ≤ a ′ , respectively. In this paper, we prove that if ℕ = A ∪ B and A ∩ B = { 8 k : k ∈ ℕ } , then R i ( A , n ) = R i ( B , n ) cannot hold for all sufficiently large integers n where i = 2 , 3 .