Arithmetic properties of exceptional lattice paths

Abstract

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