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New examples of complete sets, with connections to a Diophantine theorem of Furstenberg

Volume 177 / 2017

Vitaly Bergelson, David Simmons Acta Arithmetica 177 (2017), 101-131 MSC: Primary 11B13, 11J71. DOI: 10.4064/aa8221-10-2016 Published online: 28 December 2016


A set $A\subseteq\mathbb N$ is called complete if every sufficiently large integer can be written as a sum of distinct elements of $A$. We present a new method for proving the completeness of a set, improving results of Cassels (1960), Zannier (1992), Burr, Erdős, Graham, and Li (1996), and Hegyvári (2000). We also introduce the somewhat philosophically related notion of a dispersing set, and refine a theorem of Furstenberg (1967).


  • Vitaly BergelsonDepartment of Mathematics
    Ohio State University
    231 W. 18th Avenue
    Columbus, OH 43210-1174, U.S.A.
  • David SimmonsDepartment of Mathematics
    University of York
    Heslington, York YO10 5DD, UK

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