Three-dimensional continued fractions and Kloosterman sums

A. V. Ustinov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 70 (2015) no. 3, pp. 483-556

Résumé

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.

Détail

Keywords: three-dimensional continued fractions, lattices, Kloosterman sums, Gauss–Kuz'min statistics.

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