Multidimensional Gauss–Kuzmin Statistics
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In this chapter we study the distribution of faces in multidimensional continued fractions. The one-dimensional Gauss–Kuzmin distribution was described in Chap. 9, where we discussed the classical approach via ergodic theory and a new geometric approach. Currently, an ergodic approach to the distribution of faces in continued fractions has not been developed. In fact, it is a hard open problem to find an appropriate generalization of the Gauss map suitable to the study of ergodic properties of faces of multidimensional sails. This problem can be avoided, and in fact, the information on the distribution of faces is found via the generalization of the geometric approach via Möbius measures described in the second part of Chap. 9 for the one-dimensional case. We discuss the multidimensional analogue of the geometric approach in this chapter.
KeywordsRelative Frequency Smooth Manifold Continue Fraction Geometric Approach Integer Point
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