is called badly approximable by rational numbers if there exists
a constant C such that for all 
and 
|
If we allow to vary r but fix the dynamical system of the packing,
the construction of packings with simultaneous lower density bound has
a relation with simultaneous Diophantine approximation.
A vector
is called badly approximable by rational numbers if there exists
a constant C such that for all and
where 
. Such numbers exist (see Theorem 6F
in [12]). Denote with C(d) the maximum of all possible
constants C.
Remark.
It is trivial to get in any dimension
packings with a center-density higher than
because .
The uniform bound for the density in r gives
not dense packings and the nontriviality of
Proposition 5.1 lies in the
fact that the packings are dynamically
isospectral in the sense that they have the
same dynamical point spectrum and that we can do the packing
simultaneous for any r bigger than 0.
The estimate 
is
crude for small r.
