  The so-called Dao numbers are a sort of measure of the asymptotic behaviour
of full properties of certain product ideals in a Noetherian local ring $R$
with infinite residue field and positive depth. In this paper, we answer a
question of H. Dao on how to bound such numbers. The auxiliary tools range from
Castelnuovo-Mumford regularity of appropriate graded structures to reduction
numbers of the maximal ideal. In particular, we substantially improve previous
results (and answer questions) by the authors. As an application, we provide
new characterizations of when $R$ is regular; for instance, we show that this
holds if and only if the maximal ideal of $R$ can be generated by a
$d$-sequence (in the sense of Huneke) if and only if the third Dao number of
any (minimal) reduction of the maximal ideal vanishes.
