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1. On superintegral Kleinian sphere packings, bugs, and arithmetic groups

   https://doi.org/10.1515/crelle-2023-0004  

   Kapovich, Michael; Kontorovich, Alex (March 2023, Journal für die reine und angewandte Mathematik (Crelles Journal))

   We develop the notion of a Kleinian Sphere Packing, a generalization of“crystallographic” (Apollonian-like) sphere packings defined in [A. Kontorovich and K. Nakamura,Geometry and arithmetic of crystallographic sphere packings,Proc. Natl. Acad. Sci. USA 116 2019, 2, 436–441].Unlike crystallographic packings, Kleinian packings exist in all dimensions, as do “superintegral” such.We extend the Arithmeticity Theorem to Kleinian packings, that is, the superintegral ones come from ℚ-arithmetic lattices of simplest type.The same holds for more general objects we call Kleinian Bugs, in which the spheres need not be disjoint but can meet with dihedral angles π/m for finitely many m. We settle two questions from Kontorovich and Nakamura (2019): (i) that the Arithmeticity Theorem is in general false over number fields, and (ii)that integral packings only arise from non-uniform lattices. 

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2. Geometry and arithmetic of crystallographic sphere packings

   https://doi.org/10.1073/pnas.1721104116  

   Kontorovich, Alex; Nakamura, Kei (December 2018, Proceedings of the National Academy of Sciences)

   We introduce the notion of a “crystallographic sphere packing,” defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit an infinite family of conformally inequivalent crystallographic packings with all radii being reciprocals of integers. We then prove a result in the opposite direction: the “superintegral” ones exist only in finitely many “commensurability classes,” all in, at most, 20 dimensions. 

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3. Local-global principles in circle packings

   https://doi.org/10.1112/S0010437X19007139  

   Fuchs, Elena; Stange, Katherine E.; Zhang, Xin (June 2019, Compositio Mathematica)

   We generalize work by Bourgain and Kontorovich [ On the local-global conjecture for integral Apollonian gaskets , Invent. Math. 196 (2014), 589–650] and Zhang [ On the local-global principle for integral Apollonian 3-circle packings , J. Reine Angew. Math. 737 , (2018), 71–110], proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group $${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$$ satisfying certain conditions, where $$K$$ is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof is that $${\mathcal{A}}$$ possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in $$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$$ containing a Zariski dense subgroup of $$\operatorname{PSL}_{2}(\mathbb{Z})$$ . 

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4. The Apollonian Staircase

   https://doi.org/10.1093/imrn/rnad007  

   Rickards, James (March 2023, International Mathematics Research Notices)

   Abstract A circle of curvature $$n\in \mathbb{Z}^+$$ is a part of finitely many primitive integral Apollonian circle packings. Each such packing has a circle of minimal curvature $$-c\leq 0$$, and we study the distribution of $c/n$ across all primitive integral packings containing a circle of curvature $$n$$. As $$n\rightarrow \infty $$, the distribution is shown to tend towards a picture we name the Apollonian staircase. A consequence of the staircase is that if we choose a random circle packing containing a circle $$C$$ of curvature $$n$$, then the probability that $$C$$ is tangent to the outermost circle tends towards $$3/\pi $$. These results are found by using positive semidefinite quadratic forms to make $$\mathbb{P}^1(\mathbb{C})$$ a parameter space for (not necessarily integral) circle packings. Finally, we examine an aspect of the integral theory known as spikes. When $$n$$ is prime, the distribution of $c/n$ is extremely smooth, whereas when $$n$$ is composite, there are certain spikes that correspond to prime divisors of $$n$$ that are at most $$\sqrt{n}$$. 

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5. The Kloosterman circle method and weighted representation numbers of quadratic forms

   https://doi.org/10.1007/s40687-025-00544-4  

   Jones, Edna (August 2025, Research in the Mathematical Sciences)

   Abstract We develop a version of the Kloosterman circle method with a bump function that is used to provide asymptotics for weighted representation numbers of nonsingular integral quadratic forms. Unlike many applications of the Kloosterman circle method, we explicitly state some constants in the error terms that depend on the quadratic form. This version of the Kloosterman circle method uses Gauss sums, Kloosterman sums, Salié sums, and a principle of nonstationary phase. We briefly discuss a potential application of this version of the Kloosterman circle method to a proof of a strong asymptotic local–global principle for certain Kleinian sphere packings. 

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