Abstract :
In this paper, we introduce a class of fractal sets, which can be recursively constructed by
two sequences {nk}kP1 and {ck}kP1. 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}kP1 to make every irrational number a 2 (0, 1) correspond to only one of
the above fractal sets called a-Cantor sets, and we found that almost all a-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.
