On the values of representation functions

Abstract

For a set A of nonnegative integers, the representation functions R ₂(A, n) and R ₃(A, n) are defined as the numbers of solutions to the equation n = a + a′ with a, a′ ∈ A, a < a′ and a ⩽ a′, respectively. Let ℕ be the set of nonnegative integers. Given n ₀ > 0, it is known that there exist A, A′ ⊆ ℕ such that R ₂(A′, n) = R ₂(ℕ \ A′, n) and R ₃(A, n) = R ₃(ℕ \ A, n) for all n ⩾ n ₀. We obtain several related results. For example, we prove that: If A ⊆ ℕ such that R ₃(A, n) = R ₃(ℕ \ A, n) for all n ⩾ n ₀, then (1) for any n ⩾ n ₀ we have R ₃(A, n) = R ₃(ℕ \ A, n) > c ₁ n − c ₂, where c ₁, c ₂ are two positive constants depending only on n ₀; (2) for any $\alpha < \frac{1} {{16}}$, the set of integers n with R ₃(A, n) > αn has the density one. The answers to the four problems in Chen-Tang (2009) are affirmative. We also pose two open problems for further research.