Toward hilbert-kunz density functions in characteristic 0

For a pair (R, I), where R is a standard graded domain of dimen- sion d over an algebraically closed field of characteristic 0, and I is a graded ideal of finite colength, we prove that the existence of limp_{→∞ eHK}(R_p, I_p) is equiv- alent, for any fixed m ≥ d - 1, to the existence of lim_p→∞ l(R_p=I^[pm]_p )=p^md. This we get as a consequence of Theorem 1.1: as p →∞, the convergence of the Hilbert-Kunz (HK) density function f(R_p, I_p) is equivalent to the convergence of the truncated HK density functions f_m(R_p, I_p) (in L^∞ norm) of the mod p reductions (R_p, I_p), for any fixed m ≥ d - 1. In particular, to define the HK density function f^∞_R,I in char 0, it is enough to prove the existence of lim_p→∞ f_m(R_p, I_p), for any fixed m ≥ d - 1. This allows us to prove the existence of e^∞_HK(R, I) in many new cases, for example, when Proj R is a Segre product of curves.