Classification of horocycle orbit closures in $\mathbb{Z}$-covers

We fully describe all horocycle orbit closures in $\mathbb{Z}$-covers of compact hyperbolic surfaces. Our results rely on a careful analysis of the efficiency of all distance minimizing geodesic rays in the cover. As a corollary we obtain in this setting that all non-maximal horocycle orbit closures, while fractal, have integer Hausdorff dimension.