Consistency strength of u_κ < 2^κ at a measurable cardinal

Ascertain whether the consistency strength required to obtain a model with u_κ < 2^κ for a measurable cardinal κ exceeds the lower bound o(κ)=κ^{++}, and, if so, quantify the necessary strength.

Background

For an infinite cardinal κ≥ω, the ultrafilter number u_κ is the minimum character among uniform ultrafilters on κ. The paper discusses known techniques to force u_κ < 2^κ using iterations akin to Mathias forcing, noting that while these do not generalize directly to higher cardinals, variants have achieved u_κ < 2^κ at measurable κ starting from a supercompact cardinal.

The authors point out that violating GCH at a measurable cardinal already yields a lower bound of o(κ)=κ^{++} for the consistency strength. They explicitly state that it is completely open whether the true consistency strength must be higher than this bound.