{"paper":{"title":"Arithmetic representations of real numbers in terms of self-similar sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.DS","authors_text":"Kan Jiang, Lifeng Xi","submitted_at":"2018-08-29T11:02:18Z","abstract_excerpt":"Suppose $n\\geq 2$ and $\\mathcal{A}_{i}\\subset \\{0,1,\\cdots ,(n-1)\\}$ for $ i=1,\\cdots ,l,$ let $K_{i}=\\bigcup\\nolimits_{a\\in \\mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},\\cdots ,m_{l}\\in \\mathbb{Z}$ with $\\prod\\nolimits_{i}m_{i}\\neq 0,$ we let \\begin{equation*} S_{x}=\\left\\{ \\mathbf{(}y_{1},\\cdots ,y_{l}\\mathbf{)}:m_{1}y_{1}+\\cdots +m_{l}y_{l}=x\\text{ with }y_{i}\\in K_{i}\\text{ }\\forall i\\right\\} . \\end{equation*} In this paper, we analyze the Hausdorff dimension and Hausdorff measure of the following set \\begin{equation*} U_{r}=\\{x:\\mathbf{Card}(S_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.09724","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"}