How can I determine these point contained in Cantor set?

Question

I did an exercise is that "which of these number belong to Cantor set :$\frac{1}{249},\frac{1}{252},\frac{31}{121}$ "

I tried to divide the interval to find the range which these points contained in.But I can not also determine the point in Cantor set or not.

Can you provide some method,thank you!!!

Answer 1

As already remarked by Henning Makholm, it is enough to compute the ternary representation of such rational numbers. We have: $$ \frac{1}{249}=0.00000222\color{red}{1}00\color{red}{1}022\ldots_3$$ so $\frac{1}{249}$ does not belong to the Cantor set, but $$ \frac{1}{252}=0.00\overline{000222}_3,\qquad \frac{31}{121}=0.\overline{02022}_3 $$ both belong to it.

Answer 2

The Cantor set consists of those points in $[0,1]$ that have base-3 representations consisting solely of the digits 0 and 2.

So carry out each division in base 3 until you find the repeat, and see if you hit upon any 1s that cannot be eliminated by the rule $0.xxx1000\ldots_3 = 0.xxx0222\ldots_3$.