{"paper":{"title":"Optimal Sparsifiers for Abelian Cayley Graphs","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DS","authors_text":"Arpon Basu, Pravesh K. Kothari, Raghu Meka, Stefan Tudose","submitted_at":"2026-07-09T09:05:00Z","abstract_excerpt":"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.\n  Our proof is strongly inspired by a recent work of Reis and Rothvoss for the const"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.08261","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.08261/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}