A family of fractal sets with Hausdorff dimension 0.618

In this paper, we introduce a class of fractal sets, which can be recursively constructed by two sequences {nk}k⩾1 and {ck}k⩾1. We obtain the exact Hausdorff dimensions of these types of fractal sets using the continued fraction map. Connection of continued fraction with El Naschie's fractal spacetime is also illustrated. Furthermore, we restrict one sequence {ck}k⩾1 to make every irrational number α ∈ (0, 1) correspond to only one of the above fractal sets called α-Cantor sets, and we found that almost all α-Cantor sets possess a common Hausdorff dimension of 0.618, which is also the Hausdorff dimension of the one-dimensional random recursive Cantor set and it is the foundation stone of E-infinity fractal spacetime theory.