Let $X$ be a nonzero finite spectrum, connective say, and consider the Whitehead tower of $n$-connected covers $\dots \to X\langle n \rangle \to X\langle n-1 \rangle \to \dots \to X\langle 0 \rangle = X$.

Question: Do the spectra $X\langle n \rangle$ grown in "size" as $n$ increases?

By "size", I mean something like "number of cells" (in a minimal cell structure). As a variant, I'd be happy to understand how the Poincare series $\sum_i \dim H_i(X\langle n \rangle) z^i$ (with $\mathbb F_p$ coefficients, say) "grows", where now there are several notions of "growth" which might be relevant.

I've restricted to nonzero finite spectra $X$ to rule out bounded-above spectra, for which $X\langle n \rangle$ is eventually zero. My hunch is that with this restriction, $X\langle n \rangle$ at the very least can't decay in size too quickly.

I'd also be interested in the unstable case -- though we should probably assume that $X$ is simply-connected to rule out pathologies.

One can attain upper bounds on the grown in size of $X \langle n \rangle$ using the cofiber sequence $X \langle n \rangle \to X\langle n-1 \rangle \to \Sigma^n H\pi_n(X)$. I'm more interested in lower bounds, though.


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