Repeated iterations of trisection (Cantor sets) I'm trying to build intuition for Cantor sets by doing repeated trisections (and removing the central open interval). Below is one trisection. $$[x,y]\to\left[x,x+\frac{y-x}{3}\right]\cup\left[y-\frac{y-x}{3},y\right]$$ In the above trisection, the central interval $\left(x+\frac{y-x}{3},y-\frac{y-x}{3}\right)$ is removed. Repeated applications trisect each segment created by the prior trisection. Is there a general formula for $n$ trisections?

Assuming you start from $[0,1]$ so the first trisection removes the interval $(\frac13,\frac23)$ then the way I think about it is that $n$th step of trisections (there are $2^{n-1}$ of them) removes those real numbers which when written in ternary (i.e. base $3$) have a $1$ in the $n$th position but not in earlier positions and which are not of the form $\frac{m}{3^n}$ for some integer $m$.

So for example:

* the first trisections removes ternary numbers of the form $0.1\ldots$

* the second set of trisections removes ternary numbers of the form $0.01\ldots$ and $0.21\ldots$

* the third set of trisections removes ternary numbers of the form $0.001\ldots$, $0.021\ldots$, $0.201\ldots$, and $0.221\ldots$




apart from numbers where $\ldots$ represents recurring $0$s or recurring $2$s.