The Five Distance Theorem For An Arbitrary Norm

Abstract

The three gap theorem states that the points of the Kronecker sequence $\alpha ,2\alpha ,\dots ,N\alpha$, considered modulo one, divide the circle into intervals of at most three distinct lengths. In a two-dimensional nearest-neighbour analogue, Haynes and Marklof proved that the Kronecker sequence modulo an arbitrary unimodular lattice determines at most five distinct nearest-neighbour distances in the Euclidean norm, and that this bound is sharp. Dettmann subsequently constructed examples attaining five distinct distances for every ${\ell }_p$-norm, $1\le p\le \infty$. We prove the corresponding upper bound for every norm on ${\mathbb{R}}^2$: for every full-rank lattice $L$, every $\mathbf{\alpha }\in {\mathbb{R}}^2$, and every $N\in \mathbb{N}$, the number of distinct nearest-neighbour distances is at most five. For strictly convex norms, the proof extends the lattice-theoretic argument of Haynes and Marklof by replacing the Euclidean angular estimates with a cone lemma based on a proper Brass angular measure. The result for arbitrary norms is then obtained by a strictly convex perturbation and a limiting argument.

10 pages, 2 figures