The first question asks about the global behavior of the power function in the case of finite gaps.
Question 1. Fix a natural number $m>1.$ For each limit ordinal $\alpha,$ including $0$, let $\phi_\alpha: \omega \to \omega$ be a function such that $\phi_\alpha(0)=m, \phi_\alpha(n)>n$ and $n_1 < n_2 \implies \phi_\alpha(n_1) \leq \phi_\alpha(n_2).$ Is there a model of set theory (ZFC) in which for each limit ordinal $\alpha$ and each natural number $n, 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+\phi_\alpha(n)}$?
See Possible values for $2^{\aleph_n}$ and $2^{\aleph_\omega}$, where the case $\alpha=0$ is considered.
The second question simply asks if an Easton like theorem in the case of finite gaps exists below $\aleph_{\omega_1}$.
Question 2. For each limit ordinal $\alpha < \omega_1,$ including $0$, let $\phi_\alpha: \omega \to \omega$ be a function such that $ \phi_\alpha(n)>n$ and $n_1 < n_2 \implies \phi_\alpha(n_1) \leq \phi_\alpha(n_2).$ Is there a model of set theory (ZFC) in which for each limit ordinal $\alpha< \omega_1$ and each natural number $n, 2^{\aleph_{\alpha+n}}=\aleph_{\alpha+\phi_\alpha(n)}$?
See Two Stationary Sets with Different Gaps of the Power Function and Power function on stationry classes for partial results.
