TITLE: Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology
QUESTION [20 upvotes]: The Stone–Čech Compactification of $\mathbb{N}$ as a discrete space has been extensively studied and can be represented using ultrafilters.
Consider $X=(\mathbb{Z},\mathcal{T})$, where $\mathcal{T}$ is the Fürstenberg topology generated by arithmetic sequences. Equipped with this exotic topology, $X$ is a topological ring, metrizable, and totally disconnected. 
Since $X$ is metrizable, it is Tychonoff and the map from $X$ to its image in $\beta X$ (its compactification) is a homeomorphism. 
Has $\beta X$ been studied? Is there a straightforward description analogous to the compactification of $\mathbb{N}$ with the discrete topology?

REPLY [7 votes]: Topologically speaking $\mathbb{Z}$ with the topology mentioned above is just (homeomorphic to) the space of rational numbers. The space $\beta\mathbb{Q}$ has been studied a lot (not as much as $\beta\mathbb{N}$), for example by Eric van Douwen in Remote Points (MR link), freely available here at DMLPL.