3 problems

Let $\sG$ be the fixed primitive integral Apollonian gasket, let $\sB=\sB(\sG)$ be the set of its circle curvatures, … and call an integer represented if it lies in $\sB$. Let…

Electrostatic energy convergence conjecture. There exists a constant $b>0$ such that

Let $\mathcal{P}$ be an Apollonian gasket and let $\mathcal R\subset\mathbb C$ satisfy $\mu(\mathcal R)>0$. For $T>0$, let $F_{T,\mathcal R}(s)$ denote the pair correlation functio…