  Given an equidistributed set in the whole Euclidean space, we have
established in [1] that there exists a constant positive $C$ such that the
observability inequality of diffusion equations holds for all $T\in]0,1[$, with
an observability cost being of the form $Ce^{C/T}$. In this paper, for any
small constant $\varepsilon>0$, we prove that there exists a nontrivial
equidistributed set (in the sense that whose complementary set is unbounded),
so that the above observability cost can be improved to a fast form of
$Ce^{\varepsilon/T}$ for certain constant $C>0$. The proof is based on the
strategy used in [1], as well as an interpolation inequality for gradients of
solutions to elliptic equations obtained recently in [2].
