On the Existence of Similar Sublattices 

J. H. Conway 
Mathematics Department 
Princeton University 
Princeton, NJ 08540 

E. M. Rains and N. J. A. Sloane 
Information Sciences Research 
AT&T Shannon Lab 
Florham Park, NJ 07932-0971 

November 17, 1998; revised October 8, 1999.

A slightly different version of this paper appeared in the
Canadian Jnl. Math., {bf 51 (1999), pp. 1300-1306.

DEDICATED TO H. S. M. COXETER 

ABSTRACT

Partial answers are given to two questions.
When does a lattice Lambda contain a sublattice Lambda'
of index N that is geometrically similar to Lambda?
When is the sublattice "clean", in the sense that the boundary of the
Voronoi cells for Lambda ' do not intersect Lambda?



For the full version see
http://www.research.att.com/~njas/doc/sim.pdf (pdf) or
http://www.research.att.com/~njas/doc/sim.ps (ps)