{"paper":{"title":"On the Measure of the Midpoints of the Cantor Set in $\\mathbb{R}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Enrique Alvarado, Yunfeng Hu","submitted_at":"2017-02-22T08:37:39Z","abstract_excerpt":"In this paper, we are going to discuss the following problem: Let $T$ be a fixed set in $\\mathbb{R}^n$. And let $S$ and $B$ he two subsets in $\\mathbb{R}^n$ such that for any $x$ in $S$, there exists an $r$ such that $x+ r T$ is a subset of $B$. How small can be $B$ be if we know the size of $S$? Stein proved that for $n$ is greater than or equal to 3 and $T$ is a sphere centered at origin, then $S$ has positive measure implies $B$ has positive measure using spherical maximal operator. Later, Bourgain and Marstrand proved the similar result for $n =2$. And we found an example for why the resul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.06705","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"}