5 problems

Nearest-spacing convergence conjecture. There exists a non-negative, monotone, continuous function $H$ on $[0,\infty)$, supported away from $0$, such that

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…

Local-global conjecture. Every sufficiently large admissible number is the curvature of a circle in $\sG$; equivalently, if $n\in\sA$ and $n\gg 1$, then $n\in\sB$.

Let $N(k)$ be the number of circles in the Apollonian gasket whose curvature is at most $k$, equivalently the number of Hirst semigroup matrices whose norm is at most $k$. Let $s$…