Connectivity Threshold of Random Geometric Graphs with Cantor Distributed Vertices

For connectivity of \emph{random geometric graphs}, where there is no density for underlying distribution of the vertices, we consider $n$ i.i.d. \emph{Cantor} distributed points on $[0,1]$. We show that for this random geometric graph, the connectivity threshold $R_{n}$, converges almost surely to a constant $1-2\phi$ where $0 < \phi < 1/2$, which for the standard Cantor distribution is 1/3. We also show that $\| R_n - (1 - 2 \phi) \|_1 \sim 2 \, C(\phi) \, n^{-1/d_{\phi}}$ where $C(\phi) > 0$ is a constant and $d_{\phi} := - {\log 2}/{\log \phi}$ is the \emph{Hausdorff dimension} of the generalized Cantor set with parameter $\phi$.