Completeness properties of perturbed sequences
type: Item
instance of: scholarly article
title: Completeness properties of perturbed sequences
zbMATH Open document ID: 0469.10037
DOI: 10.1016/0022-314X(81)90036-6
author: Stefan A. Burr
author: P. Erdős
published in: Journal of Number Theory
publication date: 1981-01-01T00:00:00Z
review text: For each sequence \(S\) of positive integers, denote by \(P(S)\) the set of integers representable as a sum of distinct terms of \(S\). The sequence \(S\) is called complete of \(P(S)\) contains all large integers, entirely complete if \(P(S)\) contains all positive integers, strongly complete if \(S\) remains complete after the removal of a finite number of terms and subcomplete if \(P(S)\) contains and infinite arithmetic progression (the definition on p. 446 has ``any'' instead of ``an''). Completeness is a delicate property in the sence that it can be destroyed by the removal of a few terms of the sequence. It is shown that even the less restrictive property of subcompleteness is not very robust since any sequence can be perturbed by adding integers (possibly \(\leq 0\)) of moderate modulus to each term to yield a sequence satisfying rather mild growth conditions has a slight perturbation which is strongly complete. On the other hand it is shown that sufficiently rapidly growing sequences are not complete and that certain classes of perturbations of particular sequences are not complete. A number of open questions arising from this work are also discussed.
Mathematics Subject Classification ID: 11B99
Mathematics Subject Classification ID: 11B25
Mathematics Subject Classification ID: 11A99
zbMATH DE Number: 3735923
zbMATH Keywords: sequence of positive integers
zbMATH Keywords: sum of distinct terms
zbMATH Keywords: completeness
zbMATH Keywords: subcompleteness
zbMATH Keywords: incompleteness
reviewed by: M. M. Dodson
MaRDI profile type: Publication
cites work: Q5638406
cites work: Q3872528
cites work: On the Representation of Integers as Sums of Distinct Terms from a Fixed Sequence
cites work: On the completeness of sequences of perturbed polynomial values
cites work: Q5621399
cites work: Q5623724
cites work: Complete sequences of polynomial values
cites work: Q5340224
full work available at URL: 0022-314x(81)90036-6
OpenAlex ID: W2160453969
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description: scientific article; zbMATH DE number 3735923