{
  "abstract": "We consider the parametrised family of sets $$ E(x,y)=\\left\\{\\sum_{n=1}^{\\infty}\\frac{\\varepsilon_n}{4^n}: (\\varepsilon_n) \\in \\{0,x,y,x+y\\}^{\\mathbb{N}}\\right\\} $$ for $(x,y) \\in \\mathbb{N}^2$. This family can be viewed through three lenses: (a) as homogeneous self-similar sets; (b) as achievement sets of bi-geometric series; or (c) as the set of `rational' orthogonal projections of the four-corner Cantor set. We synthesise these three perspectives to obtain a complete topological classification of $E(x,y)$ for $(x,y) \\in \\mathbb{N}^2$. Next, we collapse this topological classification to a binary one according to whether or not $E(x,y)$ has interior. When this binary classification is visualised, it reveals a two-colour tiling $T$ of the lattice $\\mathbb N^2$, which, despite being visibly structured, turns out to be aperiodic; indeed, we prove it has no non-trivial translational symmetries. Due to the rigidity of our model, this same binary classification simultaneously captures several dichotomies. Most notably, when the family $\\{E(x,y)\\}_{(x,y) \\in \\mathbb{N}^2}$ is viewed through the theory of self-similar sets, $T$ can be seen to describe the emergence of dimension drop within the family. Finally we examine the mechanism underlying the tiling's aperiodic order. By considering the number-theoretic properties of the tiling, we characterise its substitution structure, and discover that $T$ is a factor of a substitution tiling on four ``hidden'' arithmetically defined states.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24623",
  "author_total": 2,
  "authors": [
    {
      "name": "Natalia Jurga"
    },
    {
      "name": "Dmytro Karvatskyi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dt8k1gmq6yzej.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24623",
  "fulltext_url": null,
  "id": "dt8k1gmq6yzej",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24623",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214049765",
  "pdf_url": "https://arxiv.org/pdf/2609.24623",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dt8k1gmq6yzej.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24623",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24623"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "From Dimension Drop to Aperiodic Order",
  "title_text": "From Dimension Drop to Aperiodic Order",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/dt8k1gmq6yzej",
  "venue": "arXiv",
  "versions": []
}