On additive properties of general sequences
type: Item
instance of: scholarly article
title: On additive properties of general sequences
zbMATH Open document ID: 0818.11009
DOI: 10.1016/0012-365X(94)00108-U
author: Vera T. Sós
author: András Sárközy
author: P. Erdős
published in: Discrete Mathematics
publication date: 1995-08-24T00:00:00Z
review text: The authors give a survey of their papers on additive properties of general sequences and they prove several further results on the range of additive representation functions and on difference sets. Many related unsolved problems are discussed. Let \(\mathbb{N}= \{1, 2,\dots\}\) and \(\mathbb{N}_0= \mathbb{N}\cup \{0\}\). For \(A\subseteq \mathbb{N}_0\) and \(n\in \mathbb{N}\) the number of solutions of \(n=a+ a'\); \(a,a'\in A\) is denoted by \(r_1 (A,n)\). For the additional conditions \(a\leq a'\) or \(a< a'\) this number is denoted by \(r_2 (A,n)\) or \(r_3 (A,n)\) resp. For given ranges of the representation function \(r_1\) two theorems are proved: Theorem 11. Let \(B\subseteq \mathbb{N}_0\). There exists a set \(A\subseteq \mathbb{N}_0\) such that \(B\) equals \(\{r_1 (A,n)\mid n\in \mathbb{N}\}\) if and only if either \(B= \{0,1\}\) or \(\{0,1, 2\}\subseteq B\). Theorem 12. Let \(B\subseteq \mathbb{N}_0\). There exists a set \(A\subseteq \mathbb{N}_0\) such that \(B\) equals \(\{m\in \mathbb{N}\mid m=r_1 (A,n)\) for infinitely many \(n\in \mathbb{N}\}\). Corresponding results are also stated for \(i=2\) and 3. For \(A\subseteq \mathbb{N}_0\) let \(D(A)= \{a- a'\mid a\in A\), \(a'\in A\), \(a>a'\}\). Generalizing a theorem by \textit{O. Grošek} and \textit{R. Jajcay} [J. Comb. Math. Comb. Comput. 13, 167-174 (1993; Zbl 0777.05025)], the authors show that if a set \(B\subseteq \mathbb{N}\) contains arbitrary long sequences of consecutive integers then there exists a set \(A\subseteq \mathbb{N}_0\) such that \(D(A)=B\). In contrast to sum sets it is possible that a difference set \(D(A)\) is `dense' while \(A\) is extremely `thin' because small differences \(d= a-a'\) can be formed using large elements \(a,a'\in A\). Two related results are given. The question if for a given infinite set \(B\subseteq \mathbb{N}\) the equation \(D(A)=B\) with \(0\in A\) can have a unique solution is answered positively in the case that \(B= D(A_1)\) for an infinite \(B_3\) seque…
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11B34
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 722486
zbMATH Keywords: additive bases
zbMATH Keywords: difference bases
zbMATH Keywords: \(B_k\) sequences
zbMATH Keywords: survey
zbMATH Keywords: additive representation functions
zbMATH Keywords: difference sets
zbMATH Keywords: unsolved problems
MaRDI profile type: Publication
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full work available at URL: 0012-365x(94)00108-u
OpenAlex ID: W2047057510
description: scientific article; zbMATH DE number 722486