Zooming Into the Apollonian Super-Packing

Sammanfattning

The Apollonian super-packing is a remarkable arrangement of circles on the complex plane. For almost every point $z\in\C$, there is an infinite chain of circles surrounding it. In this thesis, we study consecutive circles in these chains and the ratios of consecutive curvatures $\gamma_{n-1}(z)/\gamma_n(z)$. Our method involves introducing a measure-preserving map $T$, which is conjectured to have a certain equidistribution property. From this assumption, we find an expression for the limiting distribution of $\gamma_{n-1}(z)/\gamma_n(z)$ as $n\to\infty$.