{
  "abstract": "We show that low discrepancy is equivalent to spectral gap for directed $k$-uniform hypergraphs. For undirected hypergraphs this recovers a theorem of Lenz and Mubayi, with a considerably shorter proof. At the heart of our argument is a Frieze--Kannan type spectral regularity lemma for hypergraphs based on the variational notion of hypergraph eigenvalues by Friedman--Wigderson, which decomposes any tensor into a bounded number of rank-one tensors plus a quasi-random tensor with small top eigenvalue. This lemma may be of independent interest, and we prove it for complex-valued, not necessarily symmetric tensors. We also briefly discuss a Szemerédi-type variant and the regularization of Cayley-type hypergraphs over finite abelian groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37912",
  "author_total": 1,
  "authors": [
    {
      "name": "Hiep Han"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c2enzp6sr7pr6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37912",
  "fulltext_url": null,
  "id": "c2enzp6sr7pr6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37912",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214926667",
  "pdf_url": "https://arxiv.org/pdf/2609.37912",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c2enzp6sr7pr6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37912",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37912"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Low discrepancy and spectral gap for directed hypergraphs via spectral regularity lemma for tensors",
  "title_text": "Low discrepancy and spectral gap for directed hypergraphs via spectral regularity lemma for tensors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/c2enzp6sr7pr6",
  "venue": "arXiv",
  "versions": []
}