Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions
Karpenkov, Oleg
Original · EN
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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