On the values of representation functions (Q5919714)

Let \(\mathbb{N}\) be the set of non-negative integers. For a set \(\mathcal{A}\subset \mathbb{N}\), let \(\mathcal{R}_{2}(\mathcal{A}, n)\) and \(\mathcal{R}_{3}(\mathcal{A}, n)\) denote the number of solutions to \(a + a' = n, a, a' \in \mathcal{A}, a < a'\), and \(a \leq a'\), respectively. \textit{G. Dombi} [Acta Arith. 103, No. 2, 137--146 (2002; Zbl 1014.11009)] (resp. \textit{Y. Chen} and \textit{B. Wang} [Acta Arith. 110, No. 3, 299--303 (2003; Zbl 1032.11008)]) answered a problem of Sárközy by showing that there exist two sets \(\mathcal{A}\) and \(\mathcal{B}\) of non-negative integers with infinite symmetric difference such that for all sufficiently large integers \(n\), \(\mathcal{R}_{2}(\mathcal{A}, n) = \mathcal{R}_{2}(\mathcal{B}, n)\) (resp. \(\mathcal{R}_{3}(\mathcal{A}, n) = \mathcal{R}_{3}(\mathcal{B}, n))\). \textit{C. Sándor} [Integers 4, Paper A18, 5 p. (2004; Zbl 1135.11305)] gave precise formulations of these results. Using character, \textit{M. Tang} [Discrete Math. 308, No. 12, 2614--2616 (2008; Zbl 1162.05003)] provided a more natural proof of Sándor's results. \textit{Y. Chen} [Sci. China, Math. 54, No. 7, 1317--1331 (2011; Zbl 1236.11017)] obtained lower bounds of \(\mathcal{R}_{2}(\mathcal{A}, n)\) and \(\mathcal{R}_{3}(\mathcal{A}, n)\). Let \(\mathcal{R}_{\mathcal{A,B}}(n)\) be the number of solutions to \(a+b = n\) with \(a \in \mathcal{A}, b \in\mathcal{B}\). Let \(\mathcal{A}_{0}\) be the set of all non-negative integers \(a\) having even number of ones in their binary representation and \(\mathcal{B}_{0} = \mathbb{N} \smallsetminus \mathcal{A}_{0}\). Chen proposed a question whether there is an absolute constant \(c\) such that for any \(\mathcal{A} \subset \mathbb{N}\) and any positive integer \(N\), if \(\mathcal{R}_{2}(\mathcal{A}, n) = \mathcal{R}_{2}(\mathbb{N} \smallsetminus \mathcal{A}, n)\) for all \(n \geq 2N - 1\) and \(\mathcal{A} \neq \mathcal{A}_{0},\mathcal{B}_{0}\), then \(\mathcal{R}_{2}(\mathcal{A}, n) = \mathcal{R}_{2}(\mathbb{N} \smallsetminus \mathcal{A}, n)\) for all \(n \geq cN \). A similar question was posed for \(\mathcal{R}_{3}(A,n)\). These questions were answered by \textit{Z. Qu} [Discrete Math. 338, No. 4, 571--575 (2015; Zbl 1308.11011)].\par In this paper, the lower bounds of results obtained by Chen have been improved and it is shown that if \(\mathcal{A}\) is a subset of \(\mathbb{N}\) and \(N\) is a positive integer such that \(\mathcal{R}_{2}(\mathcal{A}, n) = \mathcal{R}_{2}(\mathbb{N} \smallsetminus \mathcal{A}, n)\) for all \(n \geq 2N - 1\), then for any \(\varepsilon > 0\), the set of integers \(n\) with \((\frac{1}{8}-\varepsilon) n \leq \mathcal{R}_{2}(\mathcal{A}, n) \leq (\frac{1}{8}+\varepsilon)n \) has density one. Further, if \(\mathcal{R}_{3}(\mathcal{A}, n) = \mathcal{R}_{3}(\mathbb{N} \smallsetminus \mathcal{A}, n)\) for all \(n \geq 2N - 1\), applying the same method and adjusting if necessary, one can get the same result.