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Some properties of three-dimensional Klein polyhedra. (English. Russian original) Zbl 1327.11044
Sb. Math. 206, No. 4, 510-539 (2015); translation from Mat. Sb. 206, No. 4, 35-66 (2015).
In this paper the author is dealing with three-dimensional continued fractions in the sense of Klein. Recall that Klein polyhedra are convex polyhedral surfaces whose all coordinates of all vertices are integers. Let $$L_s(N)$$ be the set of integer $$s$$-dimensional lattices with determinant $$N$$. In this paper the author derives the first term of the asymptotic formula (while $$N$$ tends to infinity) for an average number of integer edges in Klein polyhedra with a fixed integer length (say, $$k$$) contained in lattices of $$L_s(N)$$.

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