The F-rational signature and drops in the Hilbert-Kunz multiplicity

Let (R, m) be a Noetherian local ring of prime characteristic p. We define the F-rational signature of R, denoted by r(R), as the infimum, taken over pairs of ideals I (sic) J such that I is generated by a system of parameters and J is a strictly larger ideal, of the drops e(HK)(I, R)) - e(HK)(J, R) in the Hilbert-Kunz multiplicity. If R is excellent, then R is F-rational if and only if r(R) > 0. The proof of this fact depends on the following result in the sequel: Given an m-primary ideal I in R, there exists a positive delta(I) is an element of R+ such that, for any ideal J (sic) I, e(HK)(I, R) - e(HK)(J, R) is either 0 or at least delta(I). We study how the F-rational signature behaves under deformation, flat local ring extension, and localization.