Dao numbers and the asymptotic behaviour of fullness

In the present paper, we study the Dao numbers ${\mathfrak{d}}_1(I),{\mathfrak{d}}_2(I)$ and ${\mathfrak{d}}_3(I)$ of an ideal $I$ of a Noetherian local ring $(R,\mathfrak{m},K)$ or a standard graded Noetherian $K$-algebra. They are defined as the smallest $\mathrm{\ell }\ge 0$ such that $I{\mathfrak{m}}^k$ is $\mathfrak{m}$-full, full, weakly $\mathfrak{m}$-full, respectively, for all $k\ge \mathrm{\ell }$. We provide general bounds for the Dao numbers in terms of the Castelnuovo-Mumford regularity of certain modules over the Rees algebra $\mathcal{R}(\mathfrak{m})$. If $R$ is a Koszul algebra, we prove that the Dao numbers are less or equal to ${\text{reg}}_{{\text{gr}}_{\mathfrak{m}}(R)}{\text{gr}}_{\mathfrak{m}}(I)$, where ${\text{gr}}_{\mathfrak{m}}(I)$ is the associated graded module of $I$. Finally, for monomial ideals, we combinatorially bound the Dao numbers in terms of asymptotic linear quotients and bounding multidegrees.