This article is cited in 2 scientific papers (total in 2 papers)
Three-dimensional continued fractions and Kloosterman sums
A. V. Ustinov
Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences
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 $\det X=P$, where $X$ is a $3\times 3$ matrix with independent coefficients and $P$ 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.
Bibliography: 127 titles.
three-dimensional continued fractions, lattices, Kloosterman sums, Gauss–Kuz'min statistics.
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Russian Mathematical Surveys, 2015, 70:3, 483–556
MSC: Primary 11-02, 11J70; Secondary 11K50, 11L05
A. V. Ustinov, “Three-dimensional continued fractions and Kloosterman sums”, Uspekhi Mat. Nauk, 70:3(423) (2015), 107–180; Russian Math. Surveys, 70:3 (2015), 483–556
Citation in format AMSBIB
\paper Three-dimensional continued fractions and Kloosterman sums
\jour Uspekhi Mat. Nauk
\jour Russian Math. Surveys
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This publication is cited in the following articles:
A. V. Ustinov, “The distribution of solutions of a determinantal equation”, Sb. Math., 206:7 (2015), 988–1019
O. Karpenkov, A. Ustinov, “Geometry and combinatoric of Minkowski–Voronoi 3-dimensional continued fractions”, J. Number Theory, 176 (2017), 375–419
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