Year: 2026
Type: preprint
Abstract: The three gap theorem states that the points of the Kronecker sequence $α,2α,\ldots,Nα$, 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, $1\leq p\leq\infty$. We prove the corresponding upper bound for every norm on $\mathbb R^2$: for every full-rank lattice $L$, every $α\in\mathbb R^2$, and every $N\in\mathbb N$, the number of distinct nearest-neighbour distances is at most five. For every norm, this bound is attained for some $L$ and $α$. 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.
Source: arXiv (Cornell University)
Authors: Mironov, Nikita A., Musin, Oleg R.
Institution: The University of Texas Rio Grande Valley
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203482875