Arithmetic properties of exceptional lattice paths

Abstract

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.