One strategy for creating aperiodic sets in $\mathbf{R}$ is to take a line $L$ of irrational slope in $\mathbf{R}^2$ along with a compact window $W \subset \mathbf{R}$ which is thought of as a subset of $L^\perp.$ For simplicity, you can just make $W$ an interval containing 0. We then take any points of $\mathbf{Z}^2$ (say the standard lattice in ${\mathbf{R}^2}$) which are also contained in $L\times W$ and project them orthogonally onto $L.$ If we identify $L$ with $\mathbf{R}$ then we have constructed an aperiodic set in $\mathbf{R}$ which also has a variety of nice dynamical and analytic properties. This construction can be significantly generalized using abelian groups, but I was thinking of a slightly different generalization.

Why not do a similar construction using $\mathbf{H}^2?$ Namely, fix a uniform tiling of $\mathbf{H}^2$ and a geodesic $\gamma$ in $\mathbf{H}^2$ which descends to a geodesic on the (potentially singular) quotient surface. We would probably want to choose a geodesic which is not closed and is dense in some region of the surface. Choose some segment of a geodesic orthogonal to $\gamma$ to act like a window and project any vertices of the uniform tiling orthogonally onto $\gamma$ if they lie within the range of the window. (The region corresponding to $L \times W$ in the hyperbolic setting would look something like a banana in the Poincare disc model of $\mathbf{H}^2$.) After composing with an isometry from $\gamma$ to $\mathbf{R}$ we might get an aperiodic point set.

Is there an obvious reason why this wouldn't yield (at the very least) an aperiodic tiling of $\mathbf{R}$ with finite local complexity?