If $X$ is a finite $p$-local spectrum, then the minimal number of cells needed to construct $X$ is exactly $\dim_{\mathbb F_p} H_\ast(X,\mathbb F_p)$. Is there an analogous result in the $K(n)$-local setting?
One should be a bit careful in formulating this question. After all, the $K(n)$-local category $Sp_{K(n)}$ has a nontrivial Picard group; these are spectra $X$ with $\dim_{K(n)_\ast} K(n)_\ast(X) = 1$ which are not suspensions of the $K(n)$-local sphere spectrum $\mathbb S_{K(n)}$. There are also many possible notions of "finite". So there are several versions of this question.
First, some notation. Given a full subcategory $\mathcal C \subseteq \mathcal D$, define $\mathcal C_0 = \{0\}$, and inductively let $\mathcal C_{n+1} \subseteq \mathcal D$ be the full subcategory of objects $D \in \mathcal D$ such that there exists a cofiber sequence $C \to C' \to D$ with $C \in \mathcal C$ and $C' \in \mathcal C_n$. For $C \in \cup_{n \in \mathbb N} \mathcal C_n$, define
$$numcells_{\mathcal C}(C)$$
to be the minimal $n \in \mathbb N$ such that $C \in \mathcal C_n$; define $numcells_{\mathcal C}(X) = \infty$ if $X \not \in \cup_{n \in \mathbb N} \mathcal C_n$.
Clearly, if $X \in Sp_{K(n)}$, we have
$$numcells_{Pic(Sp_{K(n)})} (X) \geq \dim_{K(n)_\ast} K(n)_\ast(X)$$
and, more weakly,
$$numcells_{\{\Sigma^k \mathbb S_{K(n)}\}_{k \in \mathbb Z}}(X) \geq \dim_{K(n)_\ast} K(n)_\ast(X)$$
My questions are about whether equality is "always" attained.
Question:
Let $Y$ be a finite $p$-local spectrum. Then do we have $numcells_{\{\Sigma^k \mathbb S_{K(n)}\}_{k \in \mathbb Z}}(L_{K(n)} Y) = \dim_{K(n)_\ast} K(n)_\ast(Y)$?
More generally, let $X \in Sp_{K(n)}$ be such that $numcells_{\{\Sigma^k \mathbb S_{K(n)}\}_{k \in \mathbb Z}}(X)< \infty$. Then do we have $numcells_{\{\Sigma^k \mathbb S_{K(n)}\}_{k \in \mathbb Z}}(X) = \dim_{K(n)_\ast} K(n)_\ast(X)$?
In a slightly different direction, let $X \in Sp_{K(n)}$. Then do we have $numcells_{Pic(Sp_{K(n)})} (X) = \dim_{K(n)_\ast} K(n)_\ast(X)$?
I'd also be interested in these questions $T(n)$-locally (in fact, what I'm actually most interested in is the $T(n)$-local version of Question (1)) but probably such versions of the questions are much harder.
It might also be interesting to ask about $numcells_{\{\Sigma^k L_{K(n)} F(n)\}_{k \in \mathbb Z}}(X)$ where $F(n)$ is a finite type-$n$ spectrum.
