# Expression of closed intervals appeared in the $m$-th stage of construction of the Cantor set

I am reading this paper and have a question related to the following line:

Quote: Let $$D'$$ be the collection of all closed discs in the $$xy$$ plane whose diameters are the intervals in the $$x$$-axis $$\left[\frac{n+1/3}{3^m},\frac{n+4/3}{3^m}\right],$$ where $$0\leq n\leq 3^m$$ is an integer which admits a triadic expansion free from $$1$$'s;...

Now, to construct the Cantor set, we first divide $$[0,1]$$ into three closed intervals of equal length and discard the interior of the middle one, so we have two closed intervals $$C_0, C_2$$. Then in $$C_0, C_2$$ we do the same procedure: divide each of them into three closed intervals of equal length, and discard the interior of the middle one; so we have four closed intervals $$C_{0,0}, C_{0,2}, C_{2,0}, C_{2,2}$$. And so on.

Question: In the quoted line, are the diameters nothing but all closed intervals that appeared in the $$m$$-th stage of construction of the Cantor set?

No, since the interval $$[1, 3^{-m}]$$ appears in the $$m-$$th stage of the construction of the Cantor set, however, $$\frac{n + 1/3}{3^m} \ne 1$$ for any $$n, m$$ since otherwise we have $$3n + 1 = 3^{m+1}$$ which is impossible since $$3$$ does not divide $$1$$.
• Start with $m=1$, write each non-negative integer $n$ not exceeding $3^m$ in base 3, and if this representation contains no 1's, you can compute the endpoints of the interval as in your post. Set $m=2$, repeat and so on.