{
  "abstract": "We prove sharp lower bounds for the Schrödinger maximal function on the torus T^d for all dimensions d at least 2, and as a corollary obtain sharp regularity conditions for pointwise convergence of the periodic Schrödinger equation. Combined with sufficiency results established by Compaan-Lucá-Staffilani, this yields a full resolution of Carleson's problem up to endpoint for the periodic Schrödinger equation in all dimensions at least 2, and disproves the conjectured regularity condition of Miao-Yuan-Zhao, in contrast to the corresponding question in R^d. We also prove sharp estimates on the dimensions of divergence sets for the equation for high dimensional tori. Our approach uses complex multiplication on abelian varieties.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32090",
  "author_total": 2,
  "authors": [
    {
      "name": "Inbo Gottlieb Fenves"
    },
    {
      "name": "Jiahao Tan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d58zbgk54kt85.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32090",
  "fulltext_url": null,
  "id": "d58zbgk54kt85",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32090",
  "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.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214745680",
  "pdf_url": "https://arxiv.org/pdf/2609.32090",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d58zbgk54kt85.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32090",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32090"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp lower bounds for the periodic maximal Schrödinger operator in higher dimensions",
  "title_text": "Sharp lower bounds for the periodic maximal Schrödinger operator in higher dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/d58zbgk54kt85",
  "venue": "arXiv",
  "versions": []
}