Representation Functions in the Set of Nonnegative Integers

Abstract

Let ℕ be the set of all nonnegative integers. For any integers r and m, let r + mℕ = {r + mk: k ∈ ℕ}. For S ⊆ ℕ and n ∈ ℕ, let R_S(n) denote the number of solutions of the equation n = s + s′ with s, s′ ∈ S and s < s′. Let r₁, r₂, m be integers with $0 < {r}_{1} < {r}_{2} < m, \, 2 \nmid {r}_{1}$. In this paper, we prove that there exist two sets C and D with C ∪ D = ℕ and C ∩ D = (r₁ + mℕ) ∪ (r₂ + mℕ) such that R_C(n) = R_D(n) for all n ∈ ℕ if and only if there exists a positive integer l such that r₁ = 2^2l − 1, r₂ = 2^2l+1 + 2^2l − 2 and m = 2^2l+2 − 2. This solves a problem posed by the author and Pan [Proc. Edinb. Math. Soc. (2), 2025, 68(2): 655–674].

Keywords

Sárközy’s problem / representation function / Thue–Morse sequence / 11B34 / 11B83

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