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

Vitaly Bergelson, David Simmons

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A set $A\subseteq\mathbb N$ is called $complete$ if every sufficiently large integer can be written as the sum of distinct elements of $A$. In this paper we present a new method for proving the completeness of a set, improving results of Cassels ('60), Zannier ('92), Burr, Erd\H{o}s, Graham, and Li ('96), and Hegyv\'ari ('00). We also introduce the somewhat philosophically related notion of a $dispersing$ set and refine a theorem of Furstenberg ('67).
Original languageEnglish
Pages (from-to)101-131
Number of pages31
JournalActa Arithmetica
Issue number2
Early online date28 Dec 2016
Publication statusE-pub ahead of print - 28 Dec 2016

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© Instytut Matematyczny PAN, 2017.This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details.


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