Drops in joint Hilbert–Kunz multiplicities and projective equivalence

Given two m-primary ideals a and I in a Noetherian local or graded ring, we introduce the level ideals of I with respect to a as a measure of the asymptotic growth of the drop in the joint Hilbert–Kunz multiplicity, eHK(at,I)−eHK(at,(I,x)). We study the properties that must hold for two ideals a,b to determine the same level ideals for all I, and we show that this is related to projective equivalence.