On the structure of some sets which have the same representation functions
The representation function \(R_{S}(n)\) is defined for a given set \(S \subset \mathbb{N}\) as the number of solutions of the equation \(n = s + s', s < s', s, s' \in S\). For a nonnegative integer \(a\) and a set of non-negative integers \(S\) we define the sumset \(a + S\) by \(a + S = \{a + s : s \in S\}.\) Let \(A(x)\) be the number of those integers in \(A\) which are less than or equal to \(x\). \textit{G. Dombi} [Acta Arith. 103, No. 2, 137--146 (2002; Zbl 1014.11009)], constructed two sets of non-negative integers with infinite symmetric difference such that the corresponding representation functions are identical. \textit{C. Sándor} [Integers 4, Paper A18, 5 p. (2004; Zbl 1135.11305)] obtained a stronger result using generating functions. \textit{M. Tang} [Discrete Math. 308, No. 12, 2614--2616 (2008; Zbl 1162.05003)] gave a new simple proof of Sándor's result. \textit{W. Yu} and \textit{M. Tang} [Integers 12, Paper A53, 5 p. (2012; Zbl 1283.11025)] considered partitions of natural numbers whose intersection is nonempty and prove that if \( \mathbb{N} = A \cup B\) and \(A \cap B = \{4k : k \in \mathbb{N}\},\) then \(R_{A}(n) = R_{B}(n)\) cannot hold for all sufficiently large integers \(n.\) \textit{M. Tang} [Chin. Ann. Math., Ser. A 37, No. 1, 41--46 (2016; Zbl 1363.05008)] extended their result for sets \(A, B \subset \mathbb{N}\) with \( \mathbb{N} = A \cup B\) and \(A \cap B = \{mk : m,k \in \mathbb{N},k \ge 2\}.\) \textit{Y.-G. Chen} and \textit{V. F. Lev} [Integers 16, Paper A36, 4 p. (2016; Zbl 1404.11013)] constructed an infinite family of pairs of sets \(A, B \subset \mathbb{N}\) with \(R_{A} = R_{B}\) such that \(A \cup B = \mathbb{N},\) while \(A \cap B\) is an infinite arithmetic progression properly contained in \(\mathbb{N}\). Let \(\{h_{1}, h_{2}, \cdots\} (h_{1}< h_{2}< \cdots)\) be an infinite set of positive integers. Define \[ H(h_{1}, h_{2}, \cdots)=\left\{\sum_{i} \varepsilon_{i}h_{i}:\varepsilon_{i} \in\{0,1\}\right\} \] , \[ H_{0}(h_{1}, h_{2}, \cdots)=\left\{\sum_{i} \varepsilon_{i}h_{i}:\varepsilon_{i} \in\{0,1\},2 \mid \sum_{i} \varepsilon_{i}\right\}, \] \[ H_{1}(h_{1}, h_{2}, \cdots)=\left\{\sum_{i} \varepsilon_{i}h_{i}:\varepsilon_{i} \in\{0,1\},2 \nmid \sum_{i} \varepsilon_{i}\right\}. \] We call \(H(h_1, h_2, \ldots)\) a Hilbert cube In this paper, the following result has been obtained : Let \(H = H(h_1, h_2, \ldots)\) be a nondegenerate Hilbert-cube. Let \(C\) and \(D\) be two sets of nonnegative integers such that \(C \cup D = H, C \cap D =\emptyset \) and \(0 \in C.\) Then \(R_{C}(n) = R_{D}(n)\) for every positive integer \(n\) if and only if \(C = H_0(h_1, h_2, \ldots)\), \(D = H_1(h_1, h_2, \ldots).\)
scientific article; zbMATH DE number 7011060