Dao numbers and the asymptotic Behavior of fullness

In this paper, we study the Dao numbers [Formula: see text] and [Formula: see text] of an ideal [Formula: see text] of a Noetherian local ring [Formula: see text] or a standard graded Noetherian [Formula: see text]-algebra. They are defined as the smallest [Formula: see text] such that [Formula: see text] is [Formula: see text]-full, full, weakly [Formula: see text]-full, respectively, for all [Formula: see text]. We provide general bounds for the Dao numbers in terms of the Castelnuovo–Mumford regularity of certain modules over the Rees algebra [Formula: see text]. If [Formula: see text] is a Koszul algebra, we prove that the Dao numbers are less or equal to [Formula: see text], where [Formula: see text] is the associated graded module of [Formula: see text]. Finally, for monomial ideals, we combinatorially bound the Dao numbers in terms of asymptotic linear quotients and bounding multidegrees.