Lower bounds for Hilbert-Kunz multiplicities in local rings of fixed dimension

Let (R, m) be a formally unmixed local ring of positive prime characteristic and dimension d. We examine the implications of having small Hilbert-Kunz multiplicity (i.e., close to 1). In particular, we show that if R is not regular, there exists a lower bound, strictly greater than one, depending only on d, for its Hilbert-Kunz multiplicity. 1.