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Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions. (English) Zbl 1167.11025
The author develops an integer-affine classification of three-dimensional multistory, completely empty convex marked pyramids. Then this is applied to obtain the complete lists of compact two-dimensional faces of multidimensional continued fractions in the sense of Klein lying in planes at integer distances \(2, 3, 4\), to the origin. The faces are considered up to the action of the group of integer-linear transformations.
The main result has been announced in Russ. Math. Surv. 60, No. 1, 165–166 (2005); translation from Usp. Mat. Nauk 60, No. 1, 169–170 (2005; Zbl 1167.11311).

11H06 Lattices and convex bodies (number-theoretic aspects)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
11J70 Continued fractions and generalizations
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