{
  "abstract": "For cocompact Fuchsian groups $Γ\\leq \\hbox{PSL}(2, \\mathbb{R})$ we improve Selberg's bound for the error term of the counting function in the hyperbolic lattice counting problem, achieving an essentially optimal upper bound for the error term \\begin{equation*} E(X;z, w) = O(X^{1/2+\\varepsilon}) \\end{equation*} for almost every pair of points $z,w$. We extend this pointwise result for cocompact lattices acting on the $n$-dimensional real hyperbolic space $\\mathbb{H}^n$. Moreover, in the $2$-dimensional case we study the second moment of the error term of the counting function. We improve the upper bounds of Chamizo and Cherubini for almost all pairs $z,w$, but we disprove a conjecture of Phillips and Rudnick for the case of cocompact Fuchsian groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33829",
  "author_total": 2,
  "authors": [
    {
      "name": "Dimitrios Chatzakos"
    },
    {
      "name": "Panagiotis Dimakis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9grpyax1dgp2y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33829",
  "fulltext_url": null,
  "id": "9grpyax1dgp2y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33829",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214716391",
  "pdf_url": "https://arxiv.org/pdf/2609.33829",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9grpyax1dgp2y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33829",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33829"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Square root cancellation in the hyperbolic lattice counting problem over cocompact groups",
  "title_text": "Square root cancellation in the hyperbolic lattice counting problem over cocompact groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/9grpyax1dgp2y",
  "venue": "arXiv",
  "versions": []
}