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 ). Denote with C(d) the maximum of all possible constants C.
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.