{"paper":{"title":"On self-affine tiles whose boundary is a sphere","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"J\\\"org Thuswaldner, Shu-Qin Zhang","submitted_at":"2018-11-16T09:25:38Z","abstract_excerpt":"Let $M$ be a $3\\times 3$ integer matrix each of whose eigenvalues is greater than $1$ in modulus and let $\\mathcal{D}\\subset\\mathbb{Z}^3$ be a set with $|\\mathcal{D}|=|\\det M|$, called digit set. The set equation $MT = T+\\mathcal{D}$ uniquely defines a nonempty compact set $T\\subset \\mathbb{R}^3$. If $T$ has positive Lebesgue measure it is called a $3$-dimensional self-affine tile. In the present paper we study topological properties of $3$-dimensional self-affine tiles with collinear digit set, i.e., with a digit set of the form $\\mathcal{D}=\\{0,v,2v,\\ldots, (|\\det M|-1)v\\}$ for some $v\\in\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.06718","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"}