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Additive decompositions of sets with restricted prime factors

Authors: Christian Elsholtz and Adam J. Harper
Journal: Trans. Amer. Math. Soc. 367 (2015), 7403-7427
MSC (2010): Primary 11N25, 11N36, 11P70
Published electronically: November 4, 2014
MathSciNet review: 3378834
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Abstract: 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

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

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.
Article copyright: © Copyright 2014 American Mathematical Society