A power function with a fixed finite gap everywhere

Abstract

We give an application of the extender based Radin forcing to cardinal arithmetic. Assuming $\kappa$ is a large enough cardinal we construct a model satisfying $2^{\kappa} = \kappa^{+n}$ together with $2^{\lambda} = \lambda^{+n}$ for each cardinal $\lambda < \kappa$, where $0 < n < \omega$. The cofinality of $\kappa$ can be set arbitrarily or $\kappa$ can remain inaccessible. When $\kappa$ remains an inaccessible, $V_{\kappa}$ is a model of ZFC satisfying $2^{\lambda} = \lambda^{+n}$ for all cardinals $\kappa$.