{"paper":{"title":"Multiplication on self-similar sets with overlaps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Jiangwen Gu, Kan Jiang, Lifeng Xi, Li Tian, Qianqian Ye","submitted_at":"2018-07-14T09:42:50Z","abstract_excerpt":"Let $A,B\\subset\\mathbb{R}$. Define $$A\\cdot B=\\{x\\cdot y:x\\in A, y\\in B\\}.$$ In this paper, we consider the following class of self-similar sets with overlaps. Let $K$ be the attractor of the IFS $\\{f_1(x)=\\lambda x, f_2(x)=\\lambda x+c-\\lambda,f_3(x)=\\lambda x+1-\\lambda\\}$, where $f_1(I)\\cap f_2(I)\\neq \\emptyset, (f_1(I)\\cup f_2(I))\\cap f_3(I)=\\emptyset,$ and $I=[0,1]$ is the convex hull of $K$. The main result of this paper is $K\\cdot K=[0,1]$ if and only if $(1-\\lambda)^2\\leq c$.\n  Equivalently, we give a necessary and sufficient condition such that for any $u\\in[0,1]$, $u=x\\cdot y$, where $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.05368","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"}