This is a review paper dedicated to recent results on the counting problems for Apollonian circle packings. The construction of Apollonian circle packing follows form the classical Apollonius’s theorem.
{Theorem (Apollonius of Perga, 262-190 BC).} Given 3 mutually tangent circles in the plane, there exist exactly two circles tangent to all three.
The author presents a modern proof of Apollonius’s theorem. It is turn out that the counting problems of Apollonian circle packings is related to problems in dynamics and number theory for thin groups. In order to construct an Apollonian circle packing, we begin with four mutually tangent circles in the plane and keeping adding new circles tangent to three of the previous circles indefinitely.
We call an Apollonian circle packing \(\mathcal{P}\) integral if every circle in \(\mathcal{P}\) has a integral curvature. The existence of integral Apollonian circle packing is insured by Descartes’s theorem. We say that \(\mathcal{P}\) is primitive if \(\operatorname{gcd}_{C \in \mathcal{P}}\text{{curv}}(C)=1,\) where \(\text{{curv}}(C)\) is the curvature of \(C\). The circle is prime if its curvature is prime. P. Sarnak [Am. Math. Mon. 118, No. 4, 291–306 (2011; Zbl 1260.52011)] proved that there are infinitely many prime circles.
For a bounded Apollonian circle packing \(\mathcal{P}\), there are only finitely many circles of radius bigger than a given number. Hence the following counting function is well-defined for any \(T>0\): \[ N_{\mathcal{P}}(T)=\Big|\Big\{C \in \mathcal{P} : \text{{curv}}(C) \leq T\Big\}\Big|. \] For any bounded primitive integral Apollonian packing \(\mathcal{P}\), we introduce \[ \Pi_{\mathcal{P}}(T)=\Big|\Big\{\text{{prime}}~~C \in \mathcal{P} : \text{{curv}}(C) \leq T\Big\}\Big|. \] The author presents asymptotic formulas of \(N_{\mathcal{P}}(T)\) and \(\Pi_{\mathcal{P}}(T)\) as \(T\longrightarrow +\infty\).