{"paper":{"title":"On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Derong Kong, Simon Baker","submitted_at":"2018-10-16T15:36:25Z","abstract_excerpt":"Let $F=\\{\\mathbf{p}_0,\\ldots,\\mathbf{p}_n\\}$ be a collection of points in $\\mathbb{R}^d.$ The set $F$ naturally gives rise to a family of iterated function systems consisting of contractions of the form $$S_i(\\mathbf{x})=\\lambda \\mathbf{x} +(1-\\lambda)\\mathbf{p}_i,$$ where $\\lambda \\in(0,1)$. Given $F$ and $\\lambda$ it is well known that there exists a unique non-empty compact set $X$ satisfying $X=\\cup_{i=0}^n S_i(X)$. For each $\\mathbf{x} \\in X$ there exists a sequence $\\mathbf{a}\\in\\{0,\\ldots,n\\}^{\\mathbb{N}}$ satisfying $$\\mathbf{x}=\\lim_{j\\to\\infty}(S_{a_1}\\circ \\cdots \\circ S_{a_j})(\\mat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07083","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"}