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