The cardinality of the set of algebraic integers whose reciprocal has its largest coefficient (with respect to a integral basis) in a given interval.

Question

Let $\omega_1,\ldots,\omega_n$ be an integral basis for a field $K$ over $\mathbb{Q}$. Let $\mathbb{Z}_K$ be the ring of integers of $K$. For $a = \sum_{i=1}^{n} a_i \omega_i$, we write $\| a \|_{\infty} = \max_{1 \leq i \leq n} |a_i|$.

What is the cardinality of the set $\{a: a \in \mathbb{Z}_K, M \leq \| a^{-1} \|^{-1} \leq 2M\}$?

In particular, can the cardinality be bounded between $cM^n$ and $C M^n$ for some constants $0<c<C$ depending on $K$?

Note: The cardinality of the set above is the same as that of the set $\{a: 1/a \in \mathbb{Z}_K, M \leq \| a \|^{-1} \leq 2M\}$.