Suppose in constructing a set of interest we removed the mid

Suppose in constructing a set of interest, we removed the middle fths instead of the middle thirds. (First remove (2/5,3/5), then the middle fths of the two intervals that remain, etc.)
(a) When we are done, how much of the unit interval have we removed? Explain

(b) Show that the set that remains is uncountable.

Solution

Consider the set C obtained from the interval [0,1] by first removing the middle third of the interval and then removing the middle fifths of the two remaining intervals. Now iterate this process, first removing middle thirds, then removing middle fifths.

C is a fractal with fractal dimension D=(Log4)/Log(15/2).

Consider the Cantor set that we obtain as a the cartisian product of two Cantor middle-fifths set. Then the fractal dimension would be

D = log 4/log5/2 = 2 log 2/log5/2 = 2 * 0.75647... > 1.

Hence the set that remains is uncountable.