  We calculate the effective macroscopic dielectric constant $\varepsilon_a$ of
a periodic array of spherical nanocrystals (NCs) with dielectric constant
$\varepsilon$ immersed in the medium with dielectric constant $\varepsilon_m
\ll \varepsilon$. For an array of NCs with the diameter $d$ and the distance
$D$ between their centers, which are separated by the small distance $s=D-d \ll
d$ or touch each other by small facets with radius $\rho\ll d$ what is
equivalent to $s < 0$, $|s| \ll d$ we derive two analytical asymptotics of the
function $\varepsilon_a(s)$ in the limit $\varepsilon/\varepsilon_m \gg 1$.
Using the scaling hypothesis we interpolate between them near $s=0$ to obtain
new approximated function $\varepsilon_a(s)$ for $\varepsilon/\varepsilon_m \gg
1$. It agrees with existing numerical calculations for
$\varepsilon/\varepsilon_m =30$, while the standard mean-field Maxwell-Garnett
and Bruggeman approximations fail to describe percolation-like behavior of
$\varepsilon_a(s)$ near $s = 0$. We also show that in this case the charging
energy $E_c$ of a single NC in an array of touching NCs has a non-trivial
relationship to $\varepsilon_a $, namely $E_c = \alpha e^2/\varepsilon_a d$,
where $\alpha$ varies from 1.59 to 1.95 depending on the studied
three-dimensional lattices. Our approximation for $\varepsilon_a(s)$ can be
used instead of mean field Maxwell-Garnett and Bruggeman approximations to
describe percolation like transitions near $s=0$ for other material
characteristics of NC arrays, such as conductivity.
