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Uspekhi Mat. Nauk, 2015, Volume 70, Issue 3(423), Pages 107–180 (Mi umn9637)  

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

Abstract: 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.

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

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-90002
Dynasty Foundation


DOI: https://doi.org/10.4213/rm9637

Full text: PDF file (1261 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2015, 70:3, 483–556

Bibliographic databases:

Document Type: Article
UDC: 511.336+514.174.6+511.335
MSC: Primary 11-02, 11J70; Secondary 11K50, 11L05
Received: 04.12.2014

Citation: 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

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Ustinov, “The distribution of solutions of a determinantal equation”, Sb. Math., 206:7 (2015), 988–1019  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. O. Karpenkov, A. Ustinov, “Geometry and combinatoric of Minkowski–Voronoi 3-dimensional continued fractions”, J. Number Theory, 176 (2017), 375–419  crossref  mathscinet  zmath  isi  scopus
  • Успехи математических наук Russian Mathematical Surveys
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