Characterisation of Meyer sets via the Freiman--Ruzsa theorem

TL;DR

This paper demonstrates that the Freiman--Ruzsa theorem can be used to provide an alternative proof for characterising Meyer sets, which are dense subsets of Euclidean space with uniformly discrete difference sets.

Contribution

It introduces a novel approach linking additive combinatorics to the geometric structure of Meyer sets through the Freiman--Ruzsa theorem.

Findings

Alternative proof of Meyer set characterization using Freiman--Ruzsa theorem

Establishes a connection between additive combinatorics and geometric set properties

Provides insights into the structure of relatively dense subsets in Euclidean spaces

Abstract

We show that the Freiman--Ruzsa theorem, characterising finite sets with bounded doubling, leads to an alternative proof of a characterisation of Meyer sets, that is, relatively dense subsets of Euclidean spaces whose difference sets are uniformly discrete.