Additive properties of certain sets

On the values of representation functions

“…For i = 3, Chen and Wang [2] proved that the set of nonnegative integers can be partitioned into two subsets A and B with R 3 (A, n) = R 3 (B, n) for all n n 0 . In [4], Lev gave a simple common proof to the results by Dombi [3] and Chen and Wang [2]. In [5], using the generating functions, Sándor proved the precise formulations.…”

“…For a set A ⊆ N , let R 1 (A, n), R 2 (A, n) and R 3 (A, n) denote the numbers of solutions to a + a = n, a, a ∈ A, a + a = n, a, a ∈ A, a < a , a + a = n, a, a ∈ A, a a , respectively. For i ∈ {1, 2, 3}, Sárközy asked whether there are sets A and B with infinite symmetric difference such that R i (A, n) = R i (B, n) for all sufficiently large integers n. Dombi [3] proved that the answer is negative for i = 1 and positive for i = 2. For i = 3, Chen and Wang [2] proved that the set of nonnegative integers can be partitioned into two subsets A and B with R 3 (A, n) = R 3 (B, n) for all n n 0 .…”

On additive properties of two special sequences

“…For i = 1, the answer is no. For i = 2, G. Dombi [1] proved that the set N of positive integers can be partitioned into two subsets A and B such that R 2 (A, n) = R 2 (B, n) for all n ∈ N. For i = 3, G. Dombi [1] conjectured that the answer is no. For other related results, the reader is referred to [2][3][4].…”

An extension of Nathanson’s Theorem on representation functions

“…For a given infinite set A ⊂ N the representation functions R (1) h,A (n), R (2) h,A (n) and R (3) h,A (n) are defined in the following way:…”

“…For i = 1 the answer is negative (see in [3]). For i = 2 G. Dombi [3] and for i = 3 Y. G. Chen and B. Wang [2] proved that the set of nonnegative integers can be partitioned into two subsets A and B such that R (i) 2,A (n) = R (i) 2,B (n) for all n ≥ n 0 . In [6] Lev gave a common proof to the above mentioned results of Dombi [3] and Chen and Wang [2].…”