An Apollonian circle packing (ACP) is an ancient Greek construction which is made by repeatedly inscribing circles into the triangular interstices in a Descartes configuration of four mutually tangent circles. Remarkably, if the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In this paper, we compute a lower bound for the number κ ( P , X ) \kappa (P,X) of integers less than X X occurring as curvatures in a bounded integer ACP P P , and prove a conjecture of Graham, Lagarias, Mallows, Wilkes, and Yan that the ratio κ ( P , X ) / X \kappa (P,X)/X is greater than 0 0 for X X tending to infinity.

In this paper we study the length of the longest induced cycle in the unitary Cayley graph Xn = Cay(Zn; Un), where Un is the group of units in Zn. Using residues modulo the primes dividing n, we introduce a representation of the vertices that reduces the problem to a purely combinatorial question of comparing strings of symbols. This representation allows us to prove that the multiplicity of each prime dividing n, and even the value of each prime (if sufficiently large) has no effect on the length of the longest induced cycle in Xn. We also see that if n has r distinct prime divisors, Xn always contains an induced cycle of length 2 r + 2, improving the r ln r bound of Berrezbeitia and Giudici. Moreover, we extend our results for Xn to conjunctions of complete ki-partite graphs, where ki need not be finite, and also to unitary Cayley graphs on any quotient of a Dedekind domain.

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature (n − 1, 1) is "thin", namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg's theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for n F n−1 are thin.

Bounded Apollonian circle packings (ACP's) are constructed by repeatedly inscribing circles into the triangular interstices of a configuration of four mutually tangent circles, one of which is internally tangent to the other three. If the original four circles have integer curvature, all of the circles in the packing will have integer curvature as well. In [S1], Sarnak proves that there are infinitely many circles of prime curvature and infinitely many pairs of tangent circles of prime curvature in a primitive 1 integral ACP. In this paper, we give a heuristic backed up by numerical data for the number of circles of prime curvature less than x, and the number of "kissing primes," or pairs of circles of prime curvature less than x in a primitive integral ACP. We also provide experimental evidence towards a local to global principle for the curvatures in a primitive integral ACPs.

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