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The Empty Sphere Part II

Published online by Cambridge University Press:  20 November 2018

S. S. Ryshkov  and
R. M. Erdahl
S. S. Ryshkov
Affiliation:
Steklov Institute, Moscow, USSR
R. M. Erdahl
Affiliation:
Queen's University, Kingston, Ontario
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Blow up a sphere in one of the interstices of a lattice until it is held rigidly. There will be no lattice points in the interior and sufficiently many on the boundary so that their convex hull is a solid figure. Such a sphere was called an empty sphere by B. N. Delone in 1924 when he introduced his method for lattice coverings [3, 4]. The circumscribed polytope is called an L-polytope. Our interest in such matters stems from the following result [6, Theorems 2.1 and 2.3]: With a list of the L-polytopes for lattices of dimension ≦n one can give a geometrical description of the possible sets of integer solutions of

where f satisfies the following condition (in which Z denotes the integers):

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 5 , 01 October 1988 , pp. 1058 - 1073
Copyright
Copyright © Canadian Mathematical Society 1988

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