  Let $(R,\mathfrak{m},K)$ be a local ring, and let $M$ be an $R$-module of
finite length. We study asymptotic invariants, $\beta^F_i(M,R),$ defined by
twisting with Frobenius the free resolution of $M$. This family of invariants
includes the Hilbert-Kunz multiplicity
($e_{HK}(\mathfrak{m},R)=\beta^F_0(K,R)$). We discuss several properties of
these numbers that resemble the behavior of the Hilbert-Kunz multiplicity.
Furthermore, we study when the vanishing of $\beta^F_i(M,R)$ implies that $M$
has finite projective dimension. In particular, we give a complete
characterization of the vanishing of $\beta^F_i(M,R)$ for one-dimensional
rings. As a consequence of our methods, we give conditions for the
non-existence of syzygies of finite length.
