{"paper":{"title":"On the intersection of Cantor set with the unit circle and some sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Derong Kong, Kan Jiang, Wenxia Li, Zhiqiang Wang","submitted_at":"2025-07-22T12:16:02Z","abstract_excerpt":"For $\\lambda\\in(0,1/2)$ let $K_\\lambda$ be the self-similar set in $\\mathbb{R}$ generated by the iterated function system $\\{f_0(x)=\\lambda x, f_1(x)=\\lambda x+1-\\lambda \\}$. In this paper, we investigate the intersection of the unit circle $\\mathbb{S} \\subset \\mathbb{R}^2$ with the Cartesian product $K_{\\lambda} \\times K_{\\lambda}$. We prove that for $\\lambda \\in(0, 2 - \\sqrt{3}]$, the intersection is trivial, i.e., \\[ \\mathbb{S} \\cap (K_{\\lambda} \\times K_{\\lambda}) = \\{(0,1), (1,0)\\}. \\] If $\\lambda\\in [0.330384,1/2)$, then the intersection $\\mathbb{S} \\cap (K_{\\lambda} \\times K_{\\lambda})$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.16510","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.16510/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}