{
  "abstract": "Let $Γ\\subset\\operatorname{SL}_2(\\mathbb{R})\\times\\operatorname{SL}_2(\\mathbb{R})$ be an irreducible, non-uniform lattice and define the space $X = \\operatorname{SL}_2(\\mathbb{R})\\times\\operatorname{SL}_2(\\mathbb{R})/Γ$. Let $U$ be the standard horospherical subgroup in $\\operatorname{SL}_2(\\mathbb{R})\\times\\operatorname{SL}_2(\\mathbb{R})$. We show that for every $δ>0$ and $x\\in X$ with dense $U$-orbit, the $U$-orbit evaluated at almost squares, $\\{(u_{(n^{2-δ},m^{2-δ})}~:~n,m\\in\\mathbb{N}\\}\\cdot x$, is dense in $X$. The main idea of the proof is to shadow large periodic $U$-orbits and reduce the statement to a density statement within the periodic $U$-orbit, i.e. a statement for the $2$-torus $\\mathbb{T}^2$. This then follows from an effective version of Weyl's inequality to deduce the result. The effective shadowing of periodic $U$-orbits is achieved using tools from homogeneous dynamics, namely quantitative non-divergence of horospherical subgroups, recurrence under the diagonal flow and effective equidistribution of expanding horospherical subgroups. Our approach follows the strategy of the work \\cite{KR25} of \\citeauthor{KR25} who established density of almost squares of the horocycle flow for non-uniform lattices in $\\operatorname{SL}_2(\\mathbb{R})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24752",
  "author_total": 1,
  "authors": [
    {
      "name": "Konstantin Andritsch"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9757vzdzkgbvt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24752",
  "fulltext_url": null,
  "id": "9757vzdzkgbvt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24752",
  "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.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7214031884",
  "pdf_url": "https://arxiv.org/pdf/2609.24752",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9757vzdzkgbvt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24752",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24752"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Density of almost squares of horospherical orbits in non-uniform quotients of $\\operatorname{SL}_2(\\mathbb{R})\\times\\operatorname{SL}_2(\\mathbb{R})$",
  "title_text": "Density of almost squares of horospherical orbits in non-uniform quotients of SL_2(ℝ)×SL_2(ℝ)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/9757vzdzkgbvt",
  "venue": "arXiv",
  "versions": []
}