A generalization of sumsets of set of integers
type: Item
instance of: scholarly article
title: A generalization of sumsets of set of integers
zbMATH Open document ID: 1308.11006
author: Raj Kumar Mistri
author: Ram Krishna Pandey
published in: Journal of Number Theory
publication date: 2014-09-02T00:00:00Z
review text: Let \(A\) be a nonempty finite set of integers. Denote by \(k=|A|\) its cardinality. In additive number theory, for a given positive integer \(h\), one considers the following sumsets created from \(A\) : \(h A=\{x_1+\cdots+x_h:x_i\in A, 1\leq i\leq h\}\), and [for \(h\leq k\)] \(h^\land A=\{x_1+\cdots+x_h:x_i\in A, 1\leq i\leq h~,~x_i\neq x_j~\text{if}~i\neq j\}\). The best lower bounds for \(|hA|\) and \(|h^\land A|\) in terms of \(h\) and \(k,\) and the characterization of the sets \(A\) attaining these lower bounds are known. The reader can see the book by \textit{M. B. Nathanson} [Additive number theory. Inverse problems and the geometry of sumsets. New York, NY: Springer (1996; Zbl 0859.11003)] for formulation, references and proofs. The authors extend the above mentioned results by considering, for a given integer \(r\geq1,\) the sumset denoted by \(h^{(r)}A\) and defined as follows. Let \(A=\{a_0,a_1,\dots,a_{k-1}\}\). Then define \[ h^{(r)}A:=\{\sum_{i=0}^{k-1}r_ia_i~;~0\leq r_i\leq r~,~0\leq i\leq k-1~, \sum_{i=0}^{k-1}r_i=h\}~. \] The main theorem states: ``Let \(A\) be a nonempty set of \(k\) integers. Let \(r\) and \(h\) be integers such that \(1\leq r\leq h\leq rk.\) Set \(m:=\lfloor h/r\rfloor.\) Then \[ |h^{(r)}A|\geq mr(k-m)+(h-mr)(k-2m-1)+1~. \] This lower bound is best possible.'' The authors also find all sets for which equality in the preceding relation holds. The paper ends with two open problems. The first one is to generalize the above mentioned main theorem, replacing \(r\) by \(R:=(R_0,\dots,R_{k-1})\) and allowing in the sum each \(a_i\) to be repeated at most \(R_i\) times. This problem is now solved by \textit{Q.-H. Yang} and \textit{Y.-G. Chen} [``On the cardinality of general \(h\)-fold sumsets'', Eur. J. Comb. 47, 103--114 (2015; Zbl 1448.11025)].
reviewed by: Georges Grekos
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11P70
zbMATH DE Number: 6339013
zbMATH Keywords: arithmetic progression
zbMATH Keywords: \(h\)-fold sumsets
zbMATH Keywords: direct and inverse problems
MaRDI profile type: Publication
full work available at URL: j.jnt.2014.04.005
OpenAlex ID: W1981823741
cites work: Sums of Finite Sets of Integers
cites work: Inverse Theorems for Subset Sums
cites work: Q4895030
cites work: Long arithmetic progressions in sumsets: Thresholds and bounds
cites work: Finite and infinite arithmetic progressions in sumsets
cites work: Some new results on subset sums
cites work: On the structure of the sumsets
DOI: 10.1016/J.JNT.2014.04.005
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description: scientific article; zbMATH DE number 6339013