Asymptotic Hilbert–Kunz multiplicity

For a pair (M,I), where M is finitely generated graded module over a standard graded ring R of dimension d≥2, and I is a graded ideal with ℓ(R/I)<∞ and generated by elements of the same degree, we prove that lim_q→∞⁡e₁(M,I^[q])/q^d exists, where e₁(M,I^[q]) denotes the first coefficient of the Hilbert–Samuel polynomial of (M,I^[q]). We use this to get an expression for lim_k→∞⁡[e_HK(M,I^k)−e₀(M,I^k)/d!]/k^d−1, where e_HK denotes the Hilbert–Kunz multiplicity. In particular, if dim⁡ M=d then we deduce that the difference e_HK(M,I^k)−e(M,I^k)/d! grows at least as a fixed positive multiple of k^d−1 as k→∞. This is proved using ‘renormalized’ HK density functions.