American Institute of Mathematical Sciences

2019, 15: 209-236. doi: 10.3934/jmd.2019019

The local-global principle for integral Soddy sphere packings

 Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA

Received  November 08, 2017 Revised  March 23, 2019 Published  August 2019

Fund Project: 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.

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

Citation: Alex Kontorovich. The local-global principle for integral Soddy sphere packings. Journal of Modern Dynamics, 2019, 15: 209-236. doi: 10.3934/jmd.2019019
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