We show that the character spectrum (for a singular cardinal of countable cofinality) may include any prescribed set of regular cardinals between and .
Shimon Garti. Menachem Magidor. Saharon Shelah. "On the Spectrum of Characters of Ultrafilters." Notre Dame J. Formal Logic 59 (3) 371 - 379, 2018. https://doi.org/10.1215/00294527-2018-0006