On the local-global conjecture for integral Apollonian gaskets With an appendix by Peter P. Varju

• 成果类型: Article
• 署名作者: Bourgain, Jean; Kontorovich, Alex
• 署名单位: Institute for Advanced Study - USA; Yale University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-013-0475-y
• 发表日期: 2014
• 页码: 589-650
• 关键词: circle packings; random-walks; expansion; generation; geometry

摘要

We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local obstructions) integer is the curvature of some circle in the gasket.