A number theoretical question is: what is the maximal density of a
quasi-periodic sphere packing in
with spheres of radius r centered on a subset of .
Related (but not equivalent) to this question is the problem to find
in dependence of r. We know that the maximum is always taken
for rational .
This leads to the number theoretical problem to find for fixed r;SPMgt;0, the smallest cyclic group such that there exist such that for any with , the equation
has no solution in .
We also do not know if we can in all dimensions get close to the highest
densities by choosing a special sequence of 's.
In dimensions up to 6, a class
of dense packings can be obtained by
with odd r and p=p(r,d)=(r+1)/2+ q r
is the smallest integer, such that the density
is not vanishing. The densities were
converging to the highest known densities for .
Acknowledgments: I would like to thank A. Hof for discussion and literature hints.