{"paper":{"title":"Multiple representations of real numbers on self-similar sets with overlaps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.DS","authors_text":"Jiali Zhu, Kan Jiang, Li Tian, Xiaomin Ren","submitted_at":"2018-10-11T09:43:06Z","abstract_excerpt":"Let $K$ be the attractor of the following IFS\n  $$\\{f_1(x)=\\lambda x, f_2(x)=\\lambda x +c-\\lambda,f_3(x)=\\lambda x +1-\\lambda\\}, $$\n  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 results of this paper are as follows: $$\\sqrt{K}+\\sqrt{K}=[0,2]$$ if and only if $$\\sqrt{c}+1\\geq 2\\sqrt{1-\\lambda},$$ where $\\sqrt{K}+\\sqrt{K}=\\{\\sqrt{x}+\\sqrt{y}:x,y\\in K\\}$. If $c\\geq (1-\\lambda)^2$, then $$\\dfrac{K}{K}=\\left\\{\\dfrac{x}{y}:x,y\\in K, y\\neq 0\\right\\}=\\left[0,\\infty\\right).$$ As a consequence, we prove that the fol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.04930","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"}