On rings with small Hilbert-Kunz multiplicity

A result of Watanabe and Yoshida says that an unmixed local ring of positive characteristic is regular if and only if its Hilbert-Kunz multiplicity is one. We show that, for fixed $p$ and $d$, there exists a number $\epsilon(d,p) > 0$ such that for any nonregular unmixed ring $R$ its Hilbert-Kunz multiplicity is at least $1+\epsilon(d,p)$. We also show that local rings with sufficiently small Hilbert-Kunz multiplicity are Cohen-Macaulay and $F$-rational.