Artin-Mazur heights and Yobuko heights of proper log smooth schemes of Cartier type, and Hodge-Witt decompositions and Chow groups of quasi-$F$-split threefolds

In this article we prove a fundamental inequality between Artin-Mazur heights and Yobuko heights of certain proper log smooth schemes of Cartier type over a fine log scheme whose underlying scheme is the spectrum of a perfect field $\kappa$ of characteristic $p>0$. We also prove that the cohomologies of Witt-sheaves of them are finitely generated ${\cal W}(\kappa)$-modules if the Yobuko heights of them are finite. As an application, we prove that the $p$-primary torsion parts of the Chow groups of codimension $2$ of proper smooth threefolds over $\kappa$ is of finite cotype if the Yobuko heights of them are finite. These are nontrivial generalizations of results in [JR] and [J].