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 $p$ (characteristic) and $d$ (dimension), there exist a number $ϵ(d,p)>0$ such that any nonregular unmixed ring $R$ has Hilbert-Kunz multiplicity at least $1+ϵ(d,p)$. We also show that local rings with sufficiently small Hilbert-Kunz multiplicity are Cohen-Macaulay and $F$-rational.