Asymptotic vanishing conditions which force regularity in local rings of prime characteristic

Let $(R,\text{m},k)$ be a local (Noetherian) ring of positive prime characteristic $p$ and dimension $d$. Let $G_{\text{dt}}$ be a minimal resolution of the residue field $k$, and for each $i\ge 0$, let $\gothic t_i® = \lim_{e\to \8} {\length(H_i(F^e(G_\dt)))}/{p^{ed}}$\mathrm{.}Weshowthatif$\gothic t_i® = 0$ for some $i>0$, then $R$ is a regular local ring. Using the same method, we are also able to show that if $R$ is an excellent local domain and $\Tor_i^R(k,R^+) = 0$forsome$i>0$,then$R$isregular(where$R^+$ is the absolute integral closure of $R$). Both of the two results were previously known only for $i=1$ or 2 via completely different methods.