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.