I have a short question, but that keeps me stuck for a couple of days.
Let's start saying that Cantor Set is defined as:
$$C:=\left\{x\in\mathbb{R}|x=\sum_{n\in\mathbb{N}}\frac{\alpha_n}{3^n}, \alpha_n\in\{0,2\}\right\}$$
So my question: how to modify this construction of the Cantor set in order to obtain another set “full of holes” but not of measure zero? Also the new construction has to be made restricting, in some way (and here there is the problem), q-ary expression of real numbers in $[0,1]$.
Any idea? Thank you.
PS: I searched for existing questions on the subject, but I found only constructions that use infinite intersections between sets.
Something like $\{\sum\limits_n \frac{\alpha_n}{2^{2^n}} | 0 < \alpha_n < 2^{2^{n - 1}}\}$. This mean for every $k$ we require having at least one non-zero digit in positions $2^k, \ldots 2^{k + 1} - 1$.
Each such restriction multiplies measure of our set by $\left(1 - \frac{1}{2^{k + 1}}\right)$. As $\prod\limits_{k=2}^\infty \left(1 - \frac{1}{2^{k}}\right) > 0$, our set has positive measure.