Density functions for epsilon multiplicity and families of ideals

A density function for an algebraic invariant is a measurable function on (Formula presented.) which measures the invariant on an (Formula presented.) -scale. This function carries a lot more information related to the invariant without seeking extra data. It has turned out to be a useful tool, which was introduced by the third author in Trivedi [Trans. Amer. Math. Soc. 370 (2018), no. 12, 8403–8428], to study the characteristic (Formula presented.) invariant, namely Hilbert–Kunz multiplicity of a homogeneous (Formula presented.) -primary ideal. Here, we construct density functions (Formula presented.) for a Noetherian filtration (Formula presented.) of homogeneous ideals and (Formula presented.) for a filtration given by the saturated powers of a homogeneous ideal (Formula presented.) in a standard graded domain (Formula presented.). As a consequence, we get a density function (Formula presented.) for the epsilon multiplicity (Formula presented.) of a homogeneous ideal (Formula presented.) in (Formula presented.). We further show that the function (Formula presented.) is continuous everywhere except possibly at one point, and (Formula presented.) is a continuous function everywhere and is continuously differentiable except possibly at one point. As a corollary, the epsilon density function (Formula presented.) is a compactly supported continuous function on (Formula presented.) except at one point, such that (Formula presented.). All the three functions (Formula presented.), (Formula presented.) and (Formula presented.) remain invariant under passage to the integral closure of (Formula presented.). As a corollary of this theory, we observe that the ‘rescaled’ Hilbert–Samuel multiplicities of the diagonal subalgebras form a continuous family.