{"paper":{"title":"Projections of self-similar sets with no separation condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"\\'Abel Farkas","submitted_at":"2013-07-10T16:14:47Z","abstract_excerpt":"We investigate how the Hausdorff dimension and measure of a self-similar set $K\\subseteq\\mathbb{R}^{d}$ behave under linear images. This depends on the nature of the group $\\mathcal{T}$ generated by the orthogonal parts of the defining maps of $K$. We show that if $\\mathcal{T}$ is finite then every linear image of $K$ is a graph directed attractor and there exists at least one projection of $K$ such that the dimension drops under the image of the projection. In general, with no restrictions on $\\mathcal{T}$ we establish that $\\mathcal{H}^{t}(L\\circ O(K))=\\mathcal{H}^{t}(L(K))$ for every elemen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.2841","kind":"arxiv","version":4},"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"}