{
  "abstract": "We prove that for every Cayley graph $\\mathcal{G}$ over any finite abelian group $G$, there is a weighted Cayley graph with $O(\\log |G|)$ generators that is a spectral sparsifier for $\\mathcal{G}$. This bound is optimal. Applying our bound to the group $G = \\mathbb{F}_2^n$, yields, as a corollary, $O(n/\\varepsilon^2)$-sized code sparsifiers for $\\mathbb{F}_2$-linear codes, improving on the work of Khanna, Putterman and Sudan (SODA'24) who obtained a similar result with an additional $\\mathrm{polylog}(n)$ loss. Our proof is strongly inspired by a recent work of Reis and Rothvoss for the construction of $\\ell_1$-sparsifiers. Following their work, the abelian Cayley sparsification problem can be reduced to establishing a lower bound for the volume of a certain natural convex body. This volume bound follows from a short, elementary argument that relies on character symmetry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.08261",
  "author_total": 4,
  "authors": [
    {
      "name": "Arpon Basu"
    },
    {
      "name": "Pravesh K. Kothari"
    },
    {
      "name": "Raghu Meka"
    },
    {
      "name": "Stefan Tudose"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dyhtdsnamzdgg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.08261",
  "fulltext_url": null,
  "id": "dyhtdsnamzdgg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.08261",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7167902700",
  "pdf_url": "https://arxiv.org/pdf/2607.08261",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dyhtdsnamzdgg.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2607.08261",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.08261"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Optimal Sparsifiers for Abelian Cayley Graphs",
  "title_text": "Optimal Sparsifiers for Abelian Cayley Graphs",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:08Z",
  "url": "https://research.draconitic.com/paper/dyhtdsnamzdgg",
  "venue": "arXiv",
  "versions": []
}