Let $K$ be a number field with degree $n$, $B$ be a positive integer, and $\mathcal{R}_B$ be the collection of elements in the ring of integers of $K$ such that $\alpha \in \mathcal{R}_B$ if and only if $|\sigma_i(\alpha)|\le B$ for all $i$. ($\sigma_1,\cdots,\sigma_n$ are distinct embeddings of $K$ into $\mathbb{C}$.) I learned that $\alpha \in \mathcal{R}_B$ is a root of a monic polynomial with integer coefficients, whose absolute values are at most $(2B)^n$, so $\#\mathcal{R}_B\le n(2(2B)^n+1)^n$. I also heard that indeed $\#\mathcal{R}_B \sim C_K B^n$ asymptotically for a constant $C_K$, and $C_K$ is 'interesting.'
I wonder if what I wrote above is right, (if I'm right) how I can prove $\#\mathcal{R}_B \sim C_K B^n$, and why $C_K$ is interesting. Any helps will be highly appreciated.
