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Three-dimensional continued fractions and Kloosterman sums. (English. Russian original) Zbl 1332.11073
Russ. Math. Surv. 70, No. 3, 483-556 (2015); translation from Usp. Mat. Nauk. 70, No. 3, 107-180 (2015).
This survey is devoted to results related to metric properties of classical continued fractions and Voronoi-Minkowski three-dimensional continued fractions. The main focus is on applications of analytic methods based on estimates of Kloosterman sums. An apparatus is developed for solving problems about three-dimensional lattices. The approach is based on reduction to the preceding dimension, an idea used earlier by Linnik and Skubenko in the study of integer solutions of the determinant equation, where is a matrix with independent coefficients and is an increasing parameter. The proposed method is used for studying statistical properties of Voronoi-Minkowski three-dimensional continued fractions in lattices with a fixed determinant. In particular, an asymptotic formula with polynomial lowering in the remainder term is proved for the average number of Minkowski bases. This result can be regarded as a three-dimensional analogue of Porter’s theorem on the average length of finite continued fractions.

MSC:
11K50 Metric theory of continued fractions
11J70 Continued fractions and generalizations
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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