Sums of perturbed sequences of integers
type: Item
instance of: scholarly article
title: Sums of perturbed sequences of integers
zbMATH Open document ID: 0657.10059
DOI: 10.1016/0022-314X(88)90105-9
author: John R. Burke
author: William A. Webb
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: Let \(\{a_ n\}^{\infty}_{n=1}\) be an increasing sequence of nonnegative integers. Given a set P of integers, a sequence \(\{b_ n\}^{\infty}_{n=1}\) is called a P-perturbation of A if \(b_ n-a_ n\in P\) for all n. The authors investigate the stability of the notion of an asymptotic basis and a subbasis of order h under ``small'' perturbations. For instance, to every increasing sequence A of nonnegative integers, there exists a \(\{0,1,2\}\)-perturbation of A which is not a subbasis of order 2. Then the authors prove a result related to Goldbach conjecture. Namely, that there exists a \(\{-2,0,2\}\)-perturbation of the set of odd primes which is not a subbasis of order 2 and the number of perturbed primes less than n is \(\ll n/\log n \log \log n.\)
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11P32
Mathematics Subject Classification ID: 11B25
zbMATH DE Number: 4073116
zbMATH Keywords: perturbed sequences of integers
zbMATH Keywords: infinite arithmetic progression
zbMATH Keywords: stability of basis properties under small perturbations
zbMATH Keywords: asymptotic basis
zbMATH Keywords: subbasis of order h
zbMATH Keywords: Goldbach conjecture
zbMATH Keywords: number of perturbed primes
reviewed by: Štefan Porubský
MaRDI profile type: Publication
cites work: On the completeness of sequences of perturbed polynomial values
description: scientific article; zbMATH DE number 4073116