Given $a,b\in\Bbb N$, we can write $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_ba\rceil$ and $a_i<b<a$.
(1) Supposing if $b\in\mathcal{O}_K$ where $\mathcal{O}_K$ is ring of integers in number field $K$ with $1<N_{K/\Bbb Q}(b)<a$, then can we expect such a representation of $a\in\Bbb N$?
That is, is $a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$ where $t=\lceil\log_{N_{K/\Bbb Q}(b)}a\rceil$ and $N_{K/\Bbb Q}(a_i)<N_{K/\Bbb Q}(b)<a$ at every $i\in\{0,\dots,t\}$?
(2) When can we expect such representations of $\Bbb Z$ in general ring of integers $\mathcal{O}_K$ of number fields $K$?
(3) Instead of $[K:\Bbb Q]$ if we take $[K:L]$, then can we use $N_{K/L}$ instead of $N_{K/\Bbb Q}$ in some sense at least if $K,L$ are just real extensions?
(4) Fix $m\in\Bbb N$. Is there subset $S\subseteq\mathcal{O}_K\backslash\Bbb Z$ with $|S|$ where $b\in S\implies N_{K/\Bbb Q}(b)\leq\log m$ so that every $n\in\Bbb N$ with $n<m$ can be written as $$a=a_tb^t+a_{t-1}b^{t-1}+\dots+a_1b+a_0$$ where each $a_i\in\Bbb N$ for some base $b\in S$? That is, is $$\Bbb N_{\leq m}\subseteq\bigcup_{b\in S}\Bbb Z[b]?$$
Can $|S|$ be as small as $\log^c m$ where $c>0$ is fixed?
Comments:
Hurkyl's comment forbids $|S|=1$. So I am guessing $|S|=O(1)$ could be impossible as well.
$(4)$ differs from $(2)$ where $a_i\in\mathcal O_K\backslash\Bbb Z$ is allowed.