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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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