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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Additive decompositions of sets with restricted prime factors
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by Christian Elsholtz and Adam J. Harper PDF
Trans. Amer. Math. Soc. 367 (2015), 7403-7427 Request permission


We investigate sumset decompositions of quite general sets with restricted prime factors. We manage to handle certain sets, such as the smooth numbers, even though they have little sieve amenability, and conclude that these sets cannot be written as a ternary sumset. This proves a conjecture by Sárközy. We also clean up and sharpen existing results on sumset decompositions of the prime numbers.
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Additional Information
  • Christian Elsholtz
  • Affiliation: Institut für Mathematik A, Technische Universität Graz, Steyrergasse 30/II, A-8010 Graz, Austria
  • Email:
  • Adam J. Harper
  • Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, England
  • Address at time of publication: Jesus College, Cambridge CB5 8BL, England
  • MR Author ID: 871455
  • Email:
  • Received by editor(s): September 3, 2013
  • Received by editor(s) in revised form: January 28, 2014
  • Published electronically: November 4, 2014
  • Additional Notes: The first author was supported by the Austrian Science Fund (FWF): W1230
    The second author was supported by a Doctoral Prize from the Engineering and Physical Sciences Research Council of the United Kingdom, and by a postdoctoral fellowship from the Centre de recherches mathématiques in Montréal. He would also like to thank the Technische Universität Graz for their hospitality during his visit in September 2011, when the research for this paper started.
  • © Copyright 2014 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 367 (2015), 7403-7427
  • MSC (2010): Primary 11N25, 11N36, 11P70
  • DOI:
  • MathSciNet review: 3378834