{"paper":{"title":"Radial Density in Apollonian Packings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexandru Zaharescu, Cristian Cobeli, Jayadev S. Athreya","submitted_at":"2014-09-22T21:57:42Z","abstract_excerpt":"Given an Apollonian Circle Packing $\\mathcal{P}$ and a circle $C_0 = \\partial B(z_0, r_0)$ in $\\mathcal{P}$, color the set of disks in $\\mathcal{P}$ tangent to $C_0$ red. What proportion of the concentric circle $C_{\\epsilon} = \\partial B(z_0, r_0 + \\epsilon)$ is red, and what is the behavior of this quantity as $\\epsilon \\rightarrow 0$? Using equidistribution of closed horocycles on the modular surface $\\mathbb{H}^2/SL(2, \\mathbb{Z})$, we show that the answer is $\\frac{3}{\\pi} = 0.9549\\dots$ We also describe an observation due to Alex Kontorovich connecting the rate of this convergence in the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.6352","kind":"arxiv","version":2},"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"}