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Sets with Small Sumset and Rectification

Sets with Small Sumset and Rectification We study the extent to which sets A ⊆ Z / NZ , N prime, resemble sets of integers from the additive point of view (‘up to Freiman isomorphism’). We give a direct proof of a result of Freiman, namely that if | A + A | ≤ K | A | and | A | < c ( K ) N , then A is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman's structure theorem, we obtain a reasonable bound: we can take . As a byproduct of our argument we obtain a sharpening of the second author's result on sets with small sumset in torsion groups. For example, if , and if | A + A | ≤ K | A |, then A is contained in a coset of a subspace of size no more than . 2000 Mathematics Subject Classification 11B75. © London Mathematical Society « Previous | Next Article » Table of Contents This Article Bull. London Math. Soc. (2006) 38 (1): 43-52. doi: 10.1112/S0024609305018102 » Abstract Free Full Text (PDF) Free Classifications Paper Services Article metrics Alert me when cited Alert me if corrected Find similar articles Similar articles in Web of Science Add to my archive Download citation Citing Articles Load citing article information Citing articles via CrossRef Citing articles via Scopus Citing articles via Web of Science Citing articles via Google Scholar Google Scholar Articles by Green, B. Articles by Ruzsa, I. Z. Search for related content Related Content Load related web page information Share Email this article CiteULike Delicious Facebook Google+ Mendeley Twitter What's this? Search this journal: Advanced » Current Issue October 2015 47 (5) Alert me to new issues The Journal About the journal Rights & Permissions We are mobile – find out more Published on behalf of London Mathematical Society Impact Factor: 0.704 5-Yr impact factor: 0.764 Editors Peter Jørgensen Michael White View full editorial board LMS journals now available in full MathJax HTML. MathJax is an open-source JavaScript display engine that produces high-quality mathematics in all modern browsers. To learn more about MathJax, please visit their site at www.MathJax.org. For Authors Instructions to authors Including copyright assignment, and offprints order forms Alerting Services Email table of contents Email Advance Access CiteTrack XML RSS feed Corporate Services What we offer Advertising sales Reprints Supplements var taxonomies = ("SCI01480", "SCI01640", "SCI01790", "SCI02030"); Most Most Read Monstrous Moonshine Yutaka Taniyama and His Time: Very Personal Recollections A Report on Harmonic Maps Convexity and Commuting Hamiltonians The Geometries of 3-Manifolds » View all Most Read articles Most Cited Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces The Geometries of 3-Manifolds Convexity and Commuting Hamiltonians On the Number of Carmichael Numbers up to x Property T for von Neumann Algebras » View all Most Cited articles Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department. Online ISSN 1469-2120 - Print ISSN 0024-6093 Copyright © 2015 London Mathematical Society Oxford Journals Oxford University Press Site Map Privacy Policy Cookie Policy Legal Notices Frequently Asked Questions Other Oxford University Press sites: Oxford University Press Oxford Journals China Oxford Journals Japan Academic & Professional books Children's & Schools Books Dictionaries & Reference Dictionary of National Biography Digital Reference English Language Teaching Higher Education Textbooks International Education Unit Law Medicine Music Online Products & Publishing Oxford Bibliographies Online Oxford Dictionaries Online Oxford English Dictionary Oxford Language Dictionaries Online Oxford Scholarship Online Reference Rights and Permissions Resources for Retailers & Wholesalers Resources for the Healthcare Industry Very Short Introductions World's Classics var gaJsHost = (("https:" == document.location.protocol) ? "https://ssl." : "http://www."); document.write(unescape("%3Cscript src='" + gaJsHost + "google-analytics.com/ga.js' type='text/javascript'%3E%3C/script%3E")); try { var pageTracker = _gat._getTracker("UA-189672-16"); pageTracker._setDomainName(".oxfordjournals.org"); pageTracker._trackPageview(); } catch(err) {} http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Sets with Small Sumset and Rectification

Sets with Small Sumset and Rectification

Bulletin of the London Mathematical Society , Volume 38 (1) – Feb 1, 2006

Abstract

We study the extent to which sets A ⊆ Z / NZ , N prime, resemble sets of integers from the additive point of view (‘up to Freiman isomorphism’). We give a direct proof of a result of Freiman, namely that if | A + A | ≤ K | A | and | A | < c ( K ) N , then A is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman's structure theorem, we obtain a reasonable bound: we can take . As a byproduct of our argument we obtain a sharpening of the second author's result on sets with small sumset in torsion groups. For example, if , and if | A + A | ≤ K | A |, then A is contained in a coset of a subspace of size no more than . 2000 Mathematics Subject Classification 11B75. © London Mathematical Society « Previous | Next Article » Table of Contents This Article Bull. London Math. Soc. (2006) 38 (1): 43-52. doi: 10.1112/S0024609305018102 » Abstract Free Full Text (PDF) Free Classifications Paper Services Article metrics Alert me when cited Alert me if corrected Find similar articles Similar articles in Web of Science Add to my archive Download citation Citing Articles Load citing article information Citing articles via CrossRef Citing articles via Scopus Citing articles via Web of Science Citing articles via Google Scholar Google Scholar Articles by Green, B. Articles by Ruzsa, I. Z. Search for related content Related Content Load related web page information Share Email this article CiteULike Delicious Facebook Google+ Mendeley Twitter What's this? Search this journal: Advanced » Current Issue October 2015 47 (5) Alert me to new issues The Journal About the journal Rights & Permissions We are mobile – find out more Published on behalf of London Mathematical Society Impact Factor: 0.704 5-Yr impact factor: 0.764 Editors Peter Jørgensen Michael White View full editorial board LMS journals now available in full MathJax HTML. MathJax is an open-source JavaScript display engine that produces high-quality mathematics in all modern browsers. To learn more about MathJax, please visit their site at www.MathJax.org. For Authors Instructions to authors Including copyright assignment, and offprints order forms Alerting Services Email table of contents Email Advance Access CiteTrack XML RSS feed Corporate Services What we offer Advertising sales Reprints Supplements var taxonomies = ("SCI01480", "SCI01640", "SCI01790", "SCI02030"); Most Most Read Monstrous Moonshine Yutaka Taniyama and His Time: Very Personal Recollections A Report on Harmonic Maps Convexity and Commuting Hamiltonians The Geometries of 3-Manifolds » View all Most Read articles Most Cited Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces The Geometries of 3-Manifolds Convexity and Commuting Hamiltonians On the Number of Carmichael Numbers up to x Property T for von Neumann Algebras » View all Most Cited articles Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department. Online ISSN 1469-2120 - Print ISSN 0024-6093 Copyright © 2015 London Mathematical Society Oxford Journals Oxford University Press Site Map Privacy Policy Cookie Policy Legal Notices Frequently Asked Questions Other Oxford University Press sites: Oxford University Press Oxford Journals China Oxford Journals Japan Academic & Professional books Children's & Schools Books Dictionaries & Reference Dictionary of National Biography Digital Reference English Language Teaching Higher Education Textbooks International Education Unit Law Medicine Music Online Products & Publishing Oxford Bibliographies Online Oxford Dictionaries Online Oxford English Dictionary Oxford Language Dictionaries Online Oxford Scholarship Online Reference Rights and Permissions Resources for Retailers & Wholesalers Resources for the Healthcare Industry Very Short Introductions World's Classics var gaJsHost = (("https:" == document.location.protocol) ? "https://ssl." : "http://www."); document.write(unescape("%3Cscript src='" + gaJsHost + "google-analytics.com/ga.js' type='text/javascript'%3E%3C/script%3E")); try { var pageTracker = _gat._getTracker("UA-189672-16"); pageTracker._setDomainName(".oxfordjournals.org"); pageTracker._trackPageview(); } catch(err) {}

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References (17)

Publisher
Oxford University Press
Copyright
Copyright © 2015 London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/S0024609305018102
Publisher site
See Article on Publisher Site

Abstract

We study the extent to which sets A ⊆ Z / NZ , N prime, resemble sets of integers from the additive point of view (‘up to Freiman isomorphism’). We give a direct proof of a result of Freiman, namely that if | A + A | ≤ K | A | and | A | < c ( K ) N , then A is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman's structure theorem, we obtain a reasonable bound: we can take . As a byproduct of our argument we obtain a sharpening of the second author's result on sets with small sumset in torsion groups. For example, if , and if | A + A | ≤ K | A |, then A is contained in a coset of a subspace of size no more than . 2000 Mathematics Subject Classification 11B75. © London Mathematical Society « Previous | Next Article » Table of Contents This Article Bull. London Math. Soc. (2006) 38 (1): 43-52. doi: 10.1112/S0024609305018102 » Abstract Free Full Text (PDF) Free Classifications Paper Services Article metrics Alert me when cited Alert me if corrected Find similar articles Similar articles in Web of Science Add to my archive Download citation Citing Articles Load citing article information Citing articles via CrossRef Citing articles via Scopus Citing articles via Web of Science Citing articles via Google Scholar Google Scholar Articles by Green, B. Articles by Ruzsa, I. Z. Search for related content Related Content Load related web page information Share Email this article CiteULike Delicious Facebook Google+ Mendeley Twitter What's this? Search this journal: Advanced » Current Issue October 2015 47 (5) Alert me to new issues The Journal About the journal Rights & Permissions We are mobile – find out more Published on behalf of London Mathematical Society Impact Factor: 0.704 5-Yr impact factor: 0.764 Editors Peter Jørgensen Michael White View full editorial board LMS journals now available in full MathJax HTML. MathJax is an open-source JavaScript display engine that produces high-quality mathematics in all modern browsers. To learn more about MathJax, please visit their site at www.MathJax.org. For Authors Instructions to authors Including copyright assignment, and offprints order forms Alerting Services Email table of contents Email Advance Access CiteTrack XML RSS feed Corporate Services What we offer Advertising sales Reprints Supplements var taxonomies = ("SCI01480", "SCI01640", "SCI01790", "SCI02030"); Most Most Read Monstrous Moonshine Yutaka Taniyama and His Time: Very Personal Recollections A Report on Harmonic Maps Convexity and Commuting Hamiltonians The Geometries of 3-Manifolds » View all Most Read articles Most Cited Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces The Geometries of 3-Manifolds Convexity and Commuting Hamiltonians On the Number of Carmichael Numbers up to x Property T for von Neumann Algebras » View all Most Cited articles Disclaimer: Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department. Online ISSN 1469-2120 - Print ISSN 0024-6093 Copyright © 2015 London Mathematical Society Oxford Journals Oxford University Press Site Map Privacy Policy Cookie Policy Legal Notices Frequently Asked Questions Other Oxford University Press sites: Oxford University Press Oxford Journals China Oxford Journals Japan Academic & Professional books Children's & Schools Books Dictionaries & Reference Dictionary of National Biography Digital Reference English Language Teaching Higher Education Textbooks International Education Unit Law Medicine Music Online Products & Publishing Oxford Bibliographies Online Oxford Dictionaries Online Oxford English Dictionary Oxford Language Dictionaries Online Oxford Scholarship Online Reference Rights and Permissions Resources for Retailers & Wholesalers Resources for the Healthcare Industry Very Short Introductions World's Classics var gaJsHost = (("https:" == document.location.protocol) ? "https://ssl." : "http://www."); document.write(unescape("%3Cscript src='" + gaJsHost + "google-analytics.com/ga.js' type='text/javascript'%3E%3C/script%3E")); try { var pageTracker = _gat._getTracker("UA-189672-16"); pageTracker._setDomainName(".oxfordjournals.org"); pageTracker._trackPageview(); } catch(err) {}

Journal

Bulletin of the London Mathematical SocietyOxford University Press

Published: Feb 1, 2006

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