Apollonian circle packings and long horopsheres in infinite volume hyperbolic 3 manifolds

Apollonian circle packings are arrangements of tangent circles that arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles.

In a recent joint work with Alex Kontorovich, we obtain the asymptotic number of circles of curvature at most T (as T tends to infinity) in any given bounded Apollonian packing.
We also obtain the optimal upper bounds for the number of primes/twin primes in any bounded “integral” Apollonian packing.

In my talk, I will try to explain these results as well as to explain how the equidistribution of long closed horospheres in infinite volume hyperbolic 3 manifolds are related.