On the structure of some sets which have the same representation functions
type: Item
instance of: scholarly article
title: On the structure of some sets which have the same representation functions
zbMATH Open document ID: 1413.11023
author: Min Tang
author: Jia-Wen Li
published in: Periodica Mathematica Hungarica
publication date: 2019-02-01T00:00:00Z
review text: 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_…
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 11B34
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 7011060
zbMATH Keywords: partition
zbMATH Keywords: representation function
MaRDI profile type: Publication
full work available at URL: s10998-018-0240-5
OpenAlex ID: W2800830545
Wikidata QID: Q129833542
DOI: 10.1007/S10998-018-0240-5
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description: scientific article; zbMATH DE number 7011060