{"paper":{"title":"Hausdorff dimension of the arithmetic sum of self-similar sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Kan Jiang","submitted_at":"2014-11-03T14:40:37Z","abstract_excerpt":"Let $\\beta>1$. We define a class of similitudes \\[S:=\\left\\{f_{i}(x)=\\dfrac{x}{\\beta^{n_i}}+a_i:n_i\\in \\mathbb{N}^{+}, a_i\\in \\mathbb{R}\\right\\}.\\] Taking any finite similitudes $\\{f_{i}(x)\\}_{i=1}^{m} $ from $S$, it is well known that there is a unique self-similar set $K_1$ satisfying $K_1=\\cup_{i=1}^{m} f_{i}(K_1)$. Similarly, another self-similar set $K_2$ can be generated via the finite contractive maps of $S$. We call $K_1+K_2=\\{x+y:x\\in K_1, y\\in K_2\\}$ the arithmetic sum of two self-similar sets. In this paper, we prove that $K_1+K_2$ is either a self-similar set or a unique attractor "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.0505","kind":"arxiv","version":3},"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"}