{"paper":{"title":"Some variants of the periodic tiling conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS","math.LO"],"primary_cat":"math.CA","authors_text":"Rachel Greenfeld, Terence Tao","submitted_at":"2025-05-10T21:15:15Z","abstract_excerpt":"The periodic tiling conjecture (PTC) asserts, for a finitely generated Abelian group $G$ and a finite subset $F$ of $G$, that if there is a set $A$ that solves the tiling equation $\\mathbb{1}_F * \\mathbb{1}_A = 1$, there is also a periodic solution $\\mathbb{1}_{A_{\\mathrm{p}}}$. This conjecture is known to hold for some groups $G$ and fail for others. In this paper we establish three variants of the PTC. The first (due to Tim Austin) replaces the constant function $1$ on the right-hand side of the tiling equation by $0$, and the indicator functions $\\mathbb{1}_F$ and $\\mathbb{1}_A$ by bounded "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.06757","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/2505.06757/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"}