The Five Distance Theorem For An Arbitrary Norm
Abstract
The three gap theorem states that the points of the Kronecker sequence α,2α,…,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 ℓp-norm, 1≤ p≤∞. We prove the corresponding upper bound for every norm on ℝ2: for every full-rank lattice L, every α∈ℝ2, and every N∈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.
Related
- The 2-Norm Distance on the Reflexive Lattice Generated by a Double Triangle Lattice of Projections
- The norm reveals projections in the positive unit sphere of an arbitrary C*-algebra
- The Hilton-Milner theorem in the ℓ2-norm
- Distance-Weighted Norm Equivalences for Analytic Functions on John Domains
- Characterizations of subdual cones via isotonicity of the norm of the metric projection and via antitonicity of angular distance
BibTeX
@misc{mironov2026five,
title = {{The Five Distance Theorem For An Arbitrary Norm}},
author = {Nikita A. Mironov and Oleg R. Musin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.12405},
eprint = {2608.12405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.12405},
}
RIS
TY - UNPB
TI - The Five Distance Theorem For An Arbitrary Norm
AU - Nikita A. Mironov
AU - Oleg R. Musin
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2608.12405
UR - https://arxiv.org/abs/2608.12405
AB - The three gap theorem states that the points of the Kronecker sequence α,2α,…,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 ℓ_p-norm, 1≤ p≤∞. We prove the corresponding upper bound for every norm on ℝ^2: for every full-rank lattice L, every α∈ℝ^2, and every N∈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.
ER -