This paper characterizes when a Delone set
$X$
in
${{\mathbb{R}}^{n}}$
is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set
$X$
, let
${{N}_{X}}\left( T \right)$
count the number of translation-inequivalent patches of radius
$T$
in
$X$
and let
${{M}_{X}}\left( T \right)$
be the minimum radius such that every closed ball of radius
${{M}_{X}}\left( T \right)$
contains the center of a patch of every one of these kinds. We show that for each of these functions there is a “gap in the spectrum” of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to
$X$
being an ideal crystal.
Explicitly, for
${{N}_{X}}\left( T \right)$
, if
$R$
is the covering radius of
$X$
then either
${{N}_{X}}\left( T \right)$
is bounded or
${{N}_{X}}\left( T \right)\,\ge \,T/2R$
for all
$T\,>\,0$
. The constant
$1/2R$
in this bound is best possible in all dimensions.
For
${{M}_{X}}\left( T \right)$
, either
${{M}_{X}}\left( T \right)$
is bounded or
${{M}_{X}}\left( T \right)\ge T/3$
for all
$T\,>\,0$
. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set
$X$
has
${{M}_{X}}\left( T \right)\,\ge \,c\left( n \right)T$
for all
$T\,>\,0$
, for a certain constant
$c\left( n \right)$
which depends on the dimension
$n$
of
$X$
and is
$>\,1/3$
when
$n\,>\,1$
.