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Publicationes Mathematicae Debrecen
Article
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arXiv.org e-Print Archive
Preprint . 2011
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Publicationes Mathematicae Debrecen
Article . 2011 . Peer-reviewed
Data sources: Crossref
https://dx.doi.org/10.48550/ar...
Article . 2011
License: arXiv Non-Exclusive Distribution
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https://dx.doi.org/10.5486/pmd...
Article
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Geometric group theory and arithmetic diameter

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Dec 2011Embargo end date: 01 Jan 2011Publisher:University of Debrecen/ Debreceni EgyetemJournal:Publicationes Mathematicae Debrecen (issn: 0033-3883, Copyright policy )
Authors: Nathanson, Melvyn B.;

doi: 10.5486/pmd.2011.5087 , 10.48550/arxiv.1101.0786

arXiv: 1101.0786

Geometric group theory and arithmetic diameter

Abstract

Let X be a group with identity e, let A be an infinite set of generators for X, and let (X,d_A) be the metric space with the word metric d_A induced by A. If the diameter of the space is infinite, then for every positive integer h there are infinitely many elements x in X with d_A(e,x)=h. It is proved that if P is a nonempty finite set of prime numbers and A is the set of positive integers whose prime factors all belong to P, then the diameter of the metric space (\Z,d_A) is infinite. Let λ_A(h) denote the smallest positive integer x with d_A(e,x)=h. It is an open problem to compute λ_A(h) and estimate its growth rate.

7 pages. Minor editorial changes and corrected typos

Keywords

Mathematics - Number Theory, Mathematics - Metric Geometry, FOS: Mathematics, Mathematics - Combinatorics, Metric Geometry (math.MG), Number Theory (math.NT), Combinatorics (math.CO), Group Theory (math.GR), 05A17, 11B05, 11B13, 11P99, 20F65, Mathematics - Group Theory

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
7
Average
Top 10%
Average
Green
bronze