Kronecker sequences beyond the torus: nearest-neighbour distances and best returns
Abstract
We study nearest-neighbour distances and strict best returns of cyclic orbits on (V× G)/Λ, where V is a finite-dimensional real normed space, G has a bi-invariant ultrametric, and Λ is an arbitrary closed subgroup. For the quotient of the maximum product metric, the number of distinct nearest-neighbour distances in every finite orbit segment is at most P(V̅)+1, where P is an open-ball packing constant and V̅ is obtained by factoring out the continuous real part of Λ. For inner-product norms the bound depends only on the span of the projected subgroup. This gives 2d+1 for the maximum norm on ℝd and strong-kissing-number bounds for inner-product norms, extending the corresponding real and adelic distance bounds. Our second main result is an exact abelianization principle: every such cyclic orbit is isometric, with its integer indexing preserved, to a cyclic orbit on an abelian mixed quotient (V× U)/Λab. The construction preserves the reduced and effective real factors. Thus noncommutativity and non-normality enlarge the class of ambient presentations but not the class of cyclic-orbit metrics. The record-count identity and its growth consequences require only an isometric embedding of a metric space. For mixed homogeneous spaces we prove packing recurrences and, using abelianization, the sharp three-step and five-step recurrences in effective Euclidean dimensions one and two, without a normalizer assumption. We also extend the purely real recurrence to noncompact quotients and give counterexamples showing why the stronger real recurrence can fail after an ultrametric factor is added.
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BibTeX
@misc{zorin2026kronecker,
title = {{Kronecker sequences beyond the torus: nearest-neighbour distances and best returns}},
author = {Evgeniy Zorin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.13027},
eprint = {2609.13027},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13027},
}
RIS
TY - UNPB
TI - Kronecker sequences beyond the torus: nearest-neighbour distances and best returns
AU - Evgeniy Zorin
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.13027
UR - https://arxiv.org/abs/2609.13027
AB - We study nearest-neighbour distances and strict best returns of cyclic orbits on (V× G)/Λ, where V is a finite-dimensional real normed space, G has a bi-invariant ultrametric, and Λ is an arbitrary closed subgroup. For the quotient of the maximum product metric, the number of distinct nearest-neighbour distances in every finite orbit segment is at most P(V̅)+1, where P is an open-ball packing constant and V̅ is obtained by factoring out the continuous real part of Λ. For inner-product norms the bound depends only on the span of the projected subgroup. This gives 2^d+1 for the maximum norm on ℝ^d and strong-kissing-number bounds for inner-product norms, extending the corresponding real and adelic distance bounds. Our second main result is an exact abelianization principle: every such cyclic orbit is isometric, with its integer indexing preserved, to a cyclic orbit on an abelian mixed quotient (V× U)/Λ_ab. The construction preserves the reduced and effective real factors. Thus noncommutativity and non-normality enlarge the class of ambient presentations but not the class of cyclic-orbit metrics. The record-count identity and its growth consequences require only an isometric embedding of a metric space. For mixed homogeneous spaces we prove packing recurrences and, using abelianization, the sharp three-step and five-step recurrences in effective Euclidean dimensions one and two, without a normalizer assumption. We also extend the purely real recurrence to noncompact quotients and give counterexamples showing why the stronger real recurrence can fail after an ultrametric factor is added.
ER -