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

TL;DR

This paper introduces a new method for proving the completeness of sets of natural numbers, improving upon previous results, and explores related concepts like dispersing sets, connecting to Furstenberg's Diophantine theorem.

Contribution

It presents a novel approach to establish set completeness and refines a classical theorem of Furstenberg, advancing the understanding of additive number theory.

Findings

New method for proving set completeness

Improved results over previous theorems

Refined understanding of Furstenberg's theorem

Abstract

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).