  We study Hodge-Tate crystals on the absolute (log-) prismatic site of
$\mathcal{O}_K$, where $\mathcal{O}_K$ is a mixed characteristic complete
discrete valuation ring with perfect residue field. We first classify
Hodge-Tate crystals by $\mathcal{O}_K$-modules equipped with certain small
endomorphisms. We then construct Sen theory over a non-Galois Kummer tower, and
use it to classify rational Hodge-Tate crystals by (log-) nearly Hodge-Tate
representations. Various cohomology comparison and vanishing results are proved
along the way.
