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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).

##### MSC:
 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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