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A number theoretical question is: what is the maximal density of a quasi-periodic sphere packing in tex2html_wrap_inline555 with spheres of radius r centered on a subset of tex2html_wrap_inline587 .

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 tex2html_wrap_inline791 .
This leads to the number theoretical problem to find for fixed r;SPMgt;0, the smallest cyclic group tex2html_wrap_inline1893 such that there exist tex2html_wrap_inline1355 such that for any tex2html_wrap_inline1357 with tex2html_wrap_inline1899 , the equation


has no solution in tex2html_wrap_inline1313 .

We also do not know if we can in all dimensions get close to the highest densities by choosing a special sequence of tex2html_wrap_inline791 's. In dimensions up to 6, a class of dense packings can be obtained by tex2html_wrap_inline1909 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.

Oliver Knill
Mon Jun 22 17:57:55 CDT 1998