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A (generalized) Klein polyhedron (KP) is defined as the convex hull of a set of lattice points inside a (convex) domain D ⊂ R. We focus on two cases where D is the interior (i) of a simplicial cone or (ii) of a paraboloid. We introduce a dynamical system, with a group action and an invariant measure, which is naturally associated with the KP, and discuss statistical properties of the boundary of a (random) KP, with respect to the invariant measure. The boundary of a KP is called a veil; it is formed by ‘faces’ that are convex polyhedra of dimension ≤ n − 1. In the case of a simplicial cone, the dynamical systems constructed are multi-dimensional generalizations of the well-known Gauss automorphism (related to the continued fraction representation of a number x ∈ (0, 1)). The case of a paraboloid is important as it emerges as the limiting case for the domains in R of the form λD, as λ→ +∞, where D is a given convex domain with a smooth boundary. In particular, we are interested in the distribution of various parameters characterising local combinatorial and geometric properties of a veil, such as the number of lattice points lying on a single face, the number of vertices of a face, etc. We find that the two cases under consideration (simplicial and parabolic) are quite different from this point of view. A by-product of our approach is that in the simplicial case it allows us to propose algorithms for searching a ‘good’ rational approximation for the components of a given vector θ; a particular algorithm of this kind was proposed earlier in [La].

Statistics of Klein polyhedra and multidimensional continued fractions