The Minkowski's question mark function (we use the sign $?$ to note this function) was designed in 1904 by Minkowski. It can be defined as an increasing bijection between $\mathbb Q$ and the set of dyadic numbers $\mathbb D$.
Surprisingly, it's also a bijection between the set of quadratic algebraic numbers $\mathbb A_2$ and $\mathbb Q$.
So we have
- $?^{-1}(\mathbb D)=\mathbb Q$
- $?^{-2}(\mathbb D)=?^{-1}(\mathbb Q)=\mathbb A_2$
Are they any characterization of the sets $?^{z}(\mathbb D)$ for each $z\in\mathbb Z$ ?
What is $\displaystyle\mathbb R\setminus\bigcup_{z\in\mathbb Z}?^z(\mathbb D)$ ?
We have $?^z(\mathbb D)\subsetneq?^{(z-1)}(\mathbb D)$ (this is fun as $?$ is a bijection from $\mathbb R$ to $\mathbb R$ and $\mathbb D$ is dense in $\mathbb R$), so $\bigcup_{z\in\mathbb Z}?^z(\mathbb D)=\lim_{z\rightarrow -\infty}?^z(\mathbb D)$
EDIT :
This question did not get any attention. If this is a known open question, please let me know!