@article{bc295eae352e4629998a243773b225de,

title = "The local-global principle for integral soddy sphere packings",

abstract = "Fix an integral Soddy sphere packing P. Let B be the set of all bends in P. A number n is called represented if n (Formula presented) B, that is, if there is a sphere in P with bend equal to n. A number n is called admissible if it is everywhere locally represented, meaning that n (Formula presented) B(mod q) for all q. It is shown that every sufficiently large admissible number is represented.",

keywords = "Arithmetic groups, Hyperbolic geometry, Local-global principle, Quadratic forms, Sphere packings, Thin groups",

author = "Alex Kontorovich",

note = "Funding Information: Received November 8, 2017; revised March 23, 2019. 2010 Mathematics Subject Classification: Primary: 11D85; Secondary: 11F06, 20H05. Key words and phrases: Sphere packings, thin groups, hyperbolic geometry, arithmetic groups, quadratic forms, local-global principle. The author is partially supported by an NSF CAREER grant DMS-1254788 and DMS-1455705, an NSF FRG grant DMS-1463940, an Alfred P. Sloan Research Fellowship, and a BSF grant. Funding Information: The author wishes to express his gratitude to Dimitri Dias, Jeff Lagarias, Yair Minsky, Kei Nakamura, Alan Reid, and Peter Sarnak for enlightening conversations, comments and corrections. Thanks also to Stony Brook University, where the bulk of this text was completed, and the referee for comments. The author is partially supported by an NSF CAREER grant DMS-1254788 and DMS-1455705, an NSF FRG grant DMS-1463940, an Alfred P. Sloan Research Fellowship, and a BSF grant.",

year = "2019",

doi = "10.3934/jmd.2019019",

language = "English (US)",

volume = "15",

pages = "209--236",

journal = "Journal of Modern Dynamics",

issn = "1930-5311",

publisher = "American Institute of Mathematical Sciences",

}