{
  "abstract": "We study nearest-neighbour distances and strict best returns of cyclic orbits on $(V\\times G)/Λ$, 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 $P(\\overline V)+1$, where $P$ is an open-ball packing constant and $\\overline V$ 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 $\\mathbb R^d$ 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 $(V\\times U)/Λ_{\\mathrm{ab}}$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13027",
  "author_total": 1,
  "authors": [
    {
      "name": "Evgeniy Zorin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ke5675bz6avj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13027",
  "fulltext_url": null,
  "id": "8ke5675bz6avj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13027",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7212816527",
  "pdf_url": "https://arxiv.org/pdf/2609.13027",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ke5675bz6avj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.13027",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13027"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Kronecker sequences beyond the torus: nearest-neighbour distances and best returns",
  "title_text": "Kronecker sequences beyond the torus: nearest-neighbour distances and best returns",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/8ke5675bz6avj",
  "venue": "arXiv",
  "versions": []
}