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dc.contributor.authorWalker, Aleden
dc.date.accessioned2019-04-16T23:30:50Z
dc.date.available2019-04-16T23:30:50Z
dc.date.issued2016-12en
dc.identifier.issn0024-6093
dc.identifier.urihttps://www.repository.cam.ac.uk/handle/1810/291728
dc.description.abstractThe classical theorem of Schnirelmann states that the primes are an additive basis for the integers. In this paper, we consider the analogous multiplicative setting of the cyclic group (β„€/𝑞β„€)× and prove a similar result. For all suitably large primes 𝑞 we define 𝑃𝜂 to be the set of primes less than 𝜂𝑞 , viewed naturally as a subset of (β„€/𝑞β„€)× . Considering the 𝑘 ‐fold product set 𝑃(𝑘)𝜂={𝑝1𝑝2β‹―𝑝𝑘:𝑝𝑖∈𝑃𝜂} , we show that, for 𝜂≫𝑞−1/4+𝜖, there exists a constant 𝑘 depending only on 𝜖 such that 𝑃(𝑘)𝜂=(β„€/𝑞β„€)× . Erdös conjectured that, for 𝜂=1, the value 𝑘=2 should suffice: although we have not been able to prove this conjecture, we do establish that 𝑃(2)1 has density at least 164(1+𝑜(1)) . We also formulate a similar theorem in almost‐primes, improving on existing results.
dc.languageenen
dc.publisherWiley
dc.rightsAll rights reserved
dc.titleA multiplicative analogue of Schnirelmann's theoremen
dc.typeArticle
prism.endingPage1028
prism.issueIdentifier6en
prism.publicationDate2016en
prism.publicationNameBulletin of the London Mathematical Societyen
prism.startingPage1018
prism.volume48en
dc.identifier.doi10.17863/CAM.38888
dcterms.dateAccepted2016-02-25en
rioxxterms.versionofrecord10.1112/blms/bdw062en
rioxxterms.versionAM
rioxxterms.licenseref.urihttp://www.rioxx.net/licenses/all-rights-reserveden
rioxxterms.licenseref.startdate2016-12en
dc.contributor.orcidWalker, Aled [0000-0002-9879-988X]
dc.identifier.eissn1469-2120
rioxxterms.typeJournal Article/Reviewen
cam.issuedOnline2016-10-25en
dc.identifier.urlhttps://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/blms/bdw062en
rioxxterms.freetoread.startdate2017-10-25
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