{
  "abstract": "Let $1,α_1,α_2$ be a basis for a cubic number field $K\\subset\\mathbb{R}$, and for $τ_1,τ_2\\ge 1$ let $G(\\mathbfα,\\mathbfτ)$ denote the number of distinct gaps between the fractional parts of the numbers $m_1α_1+m_2α_2$, with $m_1,m_2\\in\\mathbb{Z}$, $0\\le m_1<τ_1$ and $0\\le m_2<τ_2$. We show that $G(\\mathbfα,\\mathbfτ)$ is unbounded as $\\mathbfτ$ varies. When $K$ is totally real this is deduced from a density theorem of Shapira for orbits of the diagonal group on the space of lattices. When $K$ has a complex embedding we give an explicit construction, which also shows that there are constants $c,κ>0$ for which $G(\\mathbfα,\\mathbfτ)\\ge c(\\log\\max\\{τ_1,τ_2\\})^κ$ for infinitely many $\\mathbfτ\\in\\mathbb{N}^2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07471",
  "author_total": 1,
  "authors": [
    {
      "name": "Alan Haynes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cj4r00mejzbr3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07471",
  "fulltext_url": null,
  "id": "cj4r00mejzbr3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07471",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7220892231",
  "pdf_url": "https://arxiv.org/pdf/2610.07471",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cj4r00mejzbr3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07471",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07471"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Gaps theorems for linear forms with cubic coefficients",
  "title_text": "Gaps theorems for linear forms with cubic coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/cj4r00mejzbr3",
  "venue": "arXiv",
  "versions": []
}