A consistency result on long cardinal sequences

Abstract

For any regular cardinal $\kappa$ and ordinal $\eta<\kappa^{++}$it is consistent that $2^{\kappa}$ is as large as you wish, and every function $f: \eta \longrightarrow\left[\kappa, 2^{\kappa}\right] \cap$ Card with $f(\alpha)=\kappa$ for $c f(\alpha)<\kappa$ is the cardinal sequence of some locally compact scattered space.