{"paper":{"title":"Topology of planar self-affine tiles with collinear digit set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.GN","authors_text":"Beno\\^it Loridant, J\\\"org Thuswaldner, Shigeki Akiyama","submitted_at":"2018-01-09T14:43:41Z","abstract_excerpt":"We consider the self-affine tiles with collinear digit set defined as follows. Let $A,B\\in\\mathbb{Z}$ satisfy $|A|\\leq B\\geq 2$ and $M\\in\\mathbb{Z}^{2\\times2}$ be an integral matrix with characteristic polynomial $x^2+Ax+B$. Moreover, let $\\mathcal{D}=\\{0,v,2v,\\ldots,(B-1)v\\}$ for some $v\\in\\mathbb{Z}^2$ such that $v,M v$ are linearly independent. We are interested in the topological properties of the self-affine tile $\\mathcal{T}$ defined by $M\\mathcal{T}=\\bigcup_{d\\in\\mathcal{D}}(\\mathcal{T}+d)$. Lau and Leung proved that $\\mathcal{T}$ is homeomorphic to a closed disk if and only if $2|A|\\le"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.02957","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":""},"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"}