{
  "abstract": "Motivated by a theorem of A. Schmidt on the part of the complex Lagrange spectrum below $2$, we study the restricted Lagrange spectrum $L_{\\frac{1}{2}+i\\mathbb{R}}$ arising from the approximation of complex numbers of the form $\\frac{1}{2}+iα$, $α\\in\\mathbb{R}\\setminus\\mathbb{Q}$ by Gaussian rationals $p/q$ with $p,q\\in\\mathbb{Z}[i]$, $q\\neq 0$. We show that this spectrum admits a description in terms of a dynamical spectrum associated with a real horseshoe. As a consequence, we obtain several fractal properties of $L_{\\frac{1}{2}+i\\mathbb{R}}$, such as continuity of the dimension function $t\\mapsto\\dim_H(L_{\\frac{1}{2}+i\\mathbb{R}}\\cap(-\\infty,t))$. We also prove that the set of complex numbers $z$ satisfying \\begin{equation*} \\left\\lvert z-\\frac{p}{q}\\right\\rvert\\geq\\frac{1}{2|q|^2}, \\quad\\text{for all } p,q\\in\\mathbb{Z}[i], q\\neq 0, \\end{equation*} is uncountable. In fact, we show that this inequality holds for every complex number of the form $z=\\frac{1}{2}(1+iθ)$ where $θ\\in\\mathbb{R}\\setminus\\mathbb{Q}$ is a root of one of Schmidt's $C$-minimal forms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29527",
  "author_total": 4,
  "authors": [
    {
      "name": "Hao Cheng"
    },
    {
      "name": "Harold Erazo"
    },
    {
      "name": "Carlos Gustavo Moreira"
    },
    {
      "name": "Thiago Vasconcelos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8pb07c1gvpwen.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29527",
  "fulltext_url": null,
  "id": "8pb07c1gvpwen",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29527",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7204954155",
  "pdf_url": "https://arxiv.org/pdf/2608.29527",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8pb07c1gvpwen.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29527",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29527"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half",
  "title_text": "Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/8pb07c1gvpwen",
  "venue": "arXiv",
  "versions": []
}