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

### Volume 177 / 2017

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

#### Abstract

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).