F-signature and Hilbert-Kunz multiplicity: a combined approach and comparison

We present a unified approach to the study of F-signature, Hilbert-Kunz multiplicity, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question of Watanabe and Yoshida allowing the F-signature to be viewed as the infimum of relative differences in the Hilbert-Kunz multiplicities of the cofinite ideals in a local ring.