{"paper":{"title":"Rational self-affine tiles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"J\\\"org Thuswaldner, Wolfgang Steiner (LIAFA)","submitted_at":"2012-03-04T18:59:13Z","abstract_excerpt":"An integral self-affine tile is the solution of a set equation $\\mathbf{A} \\mathcal{T} = \\bigcup_{d \\in \\mathcal{D}} (\\mathcal{T} + d)$, where $\\mathbf{A}$ is an $n \\times n$ integer matrix and $\\mathcal{D}$ is a finite subset of $\\mathbb{Z}^n$. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices $\\mathbf{A} \\in \\mathbb{Q}^{n \\times n}$. We define rational self-affine tiles as compact subsets of the open subring $\\mathbb{R}^n\\times \\prod_\\mathfrak{p} K_\\mathfrak{p}$ of the ad\\'ele ring $\\mathbb{A}_K$, where the factor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.0758","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}