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 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 $2^{\aleph_\omega}$](http://www.sciencedirect.com/science/article/pii/S0168007297000377), 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 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](http://www.math.tau.ac.il/~gitik/twostationrysets8-14.pdf) and [ Power function on stationry classes](http://www.math.tau.ac.il/~gitik/powersonstat.pdf) for partial results.

Also note that in the case of non-AC, the answer to the above questions is yes, in fact more is true. See [An Easton-like Theorem for Zermelo-Fraenkel Set Theory Without Choice (Preliminary Report)](http://arxiv.org/abs/1308.1696)