# Refinement of the hierarchy theorem of a complexity class

Say $$\mathrm{CTIME}$$ is some complexity measure, syntactic or semantic (e.g. $$\mathrm{DTIME}$$ or $$\mathrm{BPTIME}$$). If we already know that for some $$f(n) = \omega(n)$$, $$\mathrm{CTIME}(n) \subsetneq \mathrm{CTIME}(f(n))$$, then, by a standard padding argument [KV87], for any $$h(n)$$ such that $$h^c(n) = \Omega(f(n))$$ with some constant $$c$$,

\begin{align*} \mathrm{CTIME}(n) \supseteq \mathrm{CTIME}(h(n)) &\implies \mathrm{CTIME}(h^i(n)) \supseteq \mathrm{CTIME}(h^{i+1}(n)) \\ & \implies \mathrm{CTIME}(n) \supseteq \mathrm{CTIME}(f(n)). \end{align*} Therefore $$\mathrm{CTIME}(n) \subsetneq \mathrm{CTIME}(h(n))$$. As an example, authors in [KV87] show that $$\mathrm{BPTIME}(n^{\log n}) \subsetneq \mathrm{BPTIME}(2^{n^\epsilon})$$ from $$\mathrm{BPTIME}(n) \subsetneq \mathrm{BPTIME}(2^{O(n)})$$.

Can we further improve this result as follows? (My intuition is NO but I don't know why. Maybe it has something to do with whether we can well order all probable "nice" functions, by defining $$\textrm{"}f \leq g\textrm{"}\ :=\ \textrm{"}f=O(g)\textrm{"}$$.)

We only need $$h$$ to satisfy that the union of "intervals" $$[\Omega(t(n)), O(h(t(n))]$$ over all time-constructable $$t(n)$$ "covers" [$$\Omega(n), O(f(n))$$].

Suppose, throughout this process, we only consider "nice" function constructed by recursively applying arithmetic operation, $$2^x$$ or $$\log x$$ (e.g. $$n^{\log n/\log \log n}$$).