Arithmetic properties of exceptional lattice paths

For a fixed real number $\rho>0$, let $L$ be an affine line of slope $\rho^{-1}$ in $\mathbb{R}^2$. We show that the closest approximation of $L$ by a path $P$ in $\mathbb{Z}^2$ is unique, except in one case, up to integral translation. We study this exceptional case. For irrational $\rho$, the projection of $P$ to $L$ yields two quasicrystallographic tilings in the sense of Lunnon and Pleasants [5]. If $\rho$ satisfies an equation $x^2=mx+1$ with $m\in\mathbb{Z}$, both quasicrystals are mapped to each other by a substitution rule. For rational $\rho$, we characterize the periodic parts of $P$ by geometric and arithmetic properties, and exhibit a relationship to the hereditary algebras $H_{\rho}(K)$ over a field $K$ introduced in a recent proof of a conjecture of Roiter.