A proof of the positive density conjecture for integer Apollonian circle packings

DOI10.1090/S0894-0347-2011-00707-8zbMath1228.11035arXiv1001.3894OpenAlexW2008002526WikidataQ123001006 ScholiaQ123001006MaRDI QIDQ3173299

Elena Fuchs, Jean Bourgain

Publication date: 27 September 2011

Published in: Journal of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1001.3894

zbMATH Keywords

circle methodquadratic formssieve methodsnumber theoryApollonian packings

Mathematics Subject Classification ID

Quadratic and bilinear Diophantine equations (11D09) General ternary and quaternary quadratic forms; forms of more than two variables (11E20) Lattice packing and covering (number-theoretic aspects) (11H31) General binary quadratic forms (11E16) Circle packings and discrete conformal geometry (52C26)

Related Items

The Apollonian structure of integer superharmonic matrices ⋮ Effective circle count for Apollonian packings and closed horospheres ⋮ Integral points on Markoff type cubic surfaces ⋮ The sensual Apollonian circle packing ⋮ Integral Apollonian circle packings and prime curvatures ⋮ The Apollonian structure of Bianchi groups ⋮ Estimates for representation numbers of binary quadratic forms and Apollonian circle packings ⋮ Apollonian packings and Kac-Moody root systems ⋮ Working session: Thin groups and super-approximation. Abstracts from the working session held October 10--15, 2021 (hybrid meeting) ⋮ On the local-global principle for integral Apollonian 3-circle packings ⋮ Strong approximation in the Apollonian group ⋮ Local-global principles in circle packings ⋮ On the local-global conjecture for integral Apollonian gaskets. With an appendix by Péter P. Varjú ⋮ Some Experiments with Integral Apollonian Circle Packings ⋮ From Apollonius to Zaremba: Local-global phenomena in thin orbits ⋮ Counting problems in Apollonian packings ⋮ Quantitative ergodic theorems and their number-theoretic applications ⋮ Apollonian circle packings: dynamics and number theory ⋮ Recursive sequences in the Ford sphere packing ⋮ The local-global principle for integral Soddy sphere packings ⋮ Spatial Statistics of Apollonian Gaskets ⋮ Expander graphs in pure and applied mathematics ⋮ On the deformation of ball packings

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• The circle method with weights for the representation of integers by quadratic forms
• Apollonian circle packings: Geometry and group theory. III: Higher dimensions
• Estimates for representation numbers of quadratic forms
• Density of integer points on affine homogeneous varieties
• Apollonian circle packings: Number theory
• Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds
• A New Application of the Hardy-Littlewood-Kloosterman Method
• A new form of the circle method, and its application to quadratic forms.
• The Sequence of Radii of the Apollonian Packing
• Some Experiments with Integral Apollonian Circle Packings
• The Apollonian Packing of Circles