Modify Cantor Set to get a compact set with empty interior and measure $0<\alpha<1$. [duplicate]

Question

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Examples of perfect sets.

The cantor set is obtained by succesively removing the "open middle third" from each closed interval in $K_n$ to obtain $K_{n+1}$, where $K_0 = [0,1]$.

The intersection of all $K_n$ results in a compact set of null measure and empty interior.

The question is about modifying this construction to get a set with empty interior (which would stem from the fact that it contains no open intervals) and measure $\alpha$, with $0<\alpha<1$.

I thought about removing a smaller interval in each step (smaller by a factor of $\alpha$), but I realised this won't work because at step $n$ the removed part is smaller by a factor of $\alpha^n$ which tends to $0$. So at each step I should remove something that is proportionally greater than what I removed at the preceding step. I suspect a series will arise.

I'll appreciate any hints that take me into the right direction.

Answer 1

An example of such a set is the fat Cantor set. A typical construction is by removing the middle $1/4^{th}$ at the first step and in general removing the middle $1/4^n$ from each of the $2^{n-1}$ intervals at the $n^{th}$ step. Hence, the measure of the set is $$1 - \sum_{n=1}^{\infty} \left( \dfrac{2^{n-1}}{4^n}\right) = 1 - \sum_{n=1}^{\infty} \left( \dfrac1{2^{n+1}}\right) = 1/2$$ and by definition it has no interval inside it.