Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions

Details Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions

2005-10-22

arxiv.org/abs/math/0510482v2

Monatshefte fuer Mathematik, vol.152, pp.217-249, 2007.

Mathematics

Number Theory

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Scientific paper

In this paper we develop an integer-affine classification of three-dimensional multistory completely empty convex marked pyramids. We apply it to obtain the complete lists of compact two-dimensional faces of multidimensional continued fractions lying in planes with integer distances to the origin equal 2, 3, 4 ... The faces are considered up to the action of the group of integer-linear transformations. In conclusion we formulate some actual unsolved problems associated with the generalizations for n-dimensional faces and more complicated face configurations.

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