The connectivity threshold of random geometric graphs with Cantor distributed vertices

For the connectivity of random geometric graphs  , where there is no density for the underlying distribution of the vertices, we consider nn i.i.d. Cantor   distributed points on [0,1][0,1]. We show that for such a random geometric graph, the connectivity threshold, RnRn, converges almost surely to a constant 1−2ϕ1−2ϕ where 0<ϕ<1/20<ϕ<1/2, which for the standard Cantor distribution is 1/3. We also show that ‖Rn−(1−2ϕ)‖1∼2C(ϕ)n−1/dϕ where C(ϕ)>0C(ϕ)>0 is a constant and dϕ≔−log2/logϕdϕ≔−log2/logϕ is the Hausdorff dimension   of the generalized Cantor set with parameter ϕϕ.