{"paper":{"title":"Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GN","authors_text":"Jun Jason Luo, King-Shun Leung","submitted_at":"2012-05-16T05:13:02Z","abstract_excerpt":"In the paper, we focus on the connectedness of planar self-affine sets $T(A,{\\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\\det (A)|=3$ and a collinear digit set ${\\mathcal{D}}=\\{0,1,b\\}v$, where $b>1$ and $v\\in {\\mathbb{R}}^2$ such that $\\{v, Av\\}$ is linearly independent. We discuss the domain of the digit $b$ to determine the connectedness of $T(A,{\\mathcal{D}})$. Especially, a complete characterization is obtained when we restrict $b$ to be an integer. Some results on the general case of $|\\det (A)|> 3$ are obtained as well."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.3552","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"}