Pair correlation in Apollonian gaskets

Let P be an Apollonian circle packing, and PT be the set of circles from P with radius >1/T. Thanks to the work of Oh-Shah, it is known that circles in PT become equidistributed according to some Hausdorff measure, as T→ ∞. In fact, more regularities hold beyond equidstribution. Wefocus on the pair correlation function PT(s)which counts pairs of nearbycircles with normalized distance < s. Using a theorem of Mohammadi-Oh, we show that limT→∞PT exists, and can be described by a continuously differentiable function. While fine scale statistics such as pair correlation have been extensively studied in mathematical physics as well as in number theory, our work is the first investigation of such statistics on fractal sets.