{"paper":{"title":"Unique expansions and intersections of Cantor sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DS","authors_text":"Derong Kong, Simon Baker","submitted_at":"2016-04-04T13:51:31Z","abstract_excerpt":"To each $\\alpha\\in(1/3,1/2)$ we associate the Cantor set $$\\Gamma_{\\alpha}:=\\Big\\{\\sum_{i=1}^{\\infty}\\epsilon_{i}\\alpha^i: \\epsilon_i\\in\\{0,1\\},\\,i\\geq 1\\Big\\}.$$\n  In this paper we consider the intersection $\\Gamma_\\alpha \\cap (\\Gamma_\\alpha + t)$ for any translation $t\\in\\mathbb{R}$. We pay special attention to those $t$ with a unique\n  $\\{-1,0,1\\}$ $\\alpha$-expansion, and study the set\n  $$D_\\alpha:=\\{\\dim_H(\\Gamma_\\alpha \\cap (\\Gamma_\\alpha + t)):t \\textrm{ has a unique }\\{-1,0,1\\}\\,\\alpha\\textrm{-expansion}\\}.$$ We prove that there exists a transcendental number $\\alpha_{KL}\\approx 0.3943"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.00858","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"}