Geodesic continued fractions and LLL

F Beukers

doi:10.1016/j.indag.2014.04.003

Geodesic continued fractions and LLL

F Beukers

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to dd real numbers α1,…,αdα1,…,αd. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter tt as t↓0t↓0. Suggestions in this direction have been made several times over in the literature, e.g. Chevallier (2013) [4] or Bosma and Smeets (2013) [2]. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in tt.

Original language	English
Pages (from-to)	632-645
Number of pages	14
Journal	Indagationes Mathematicae
Volume	25
Issue number	4
DOIs	https://doi.org/10.1016/j.indag.2014.04.003
Publication status	Published - 2014

Keywords

Multidimensional continued fraction
Minkowski reduction
LLL-reduction

Access to Document

10.1016/j.indag.2014.04.003

1-s2.0-S0019357714000329-mainFinal published version, 216 KB

Cite this

@article{be793dec030d43c8bb9835e3ad15be73,

title = "Geodesic continued fractions and LLL",

abstract = "We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to dd real numbers α1,…,αdα1,…,αd. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter tt as t↓0t↓0. Suggestions in this direction have been made several times over in the literature, e.g. Chevallier (2013) [4] or Bosma and Smeets (2013) [2]. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in tt.",

keywords = "Multidimensional continued fraction, Minkowski reduction, LLL-reduction",

author = "F Beukers",

year = "2014",

doi = "10.1016/j.indag.2014.04.003",

language = "English",

volume = "25",

pages = "632--645",

journal = "Indagationes Mathematicae",

issn = "0019-3577",

publisher = "Elsevier BV",

number = "4",

}

TY - JOUR

T1 - Geodesic continued fractions and LLL

AU - Beukers, F

PY - 2014

Y1 - 2014

N2 - We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to dd real numbers α1,…,αdα1,…,αd. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter tt as t↓0t↓0. Suggestions in this direction have been made several times over in the literature, e.g. Chevallier (2013) [4] or Bosma and Smeets (2013) [2]. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in tt.

AB - We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to dd real numbers α1,…,αdα1,…,αd. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter tt as t↓0t↓0. Suggestions in this direction have been made several times over in the literature, e.g. Chevallier (2013) [4] or Bosma and Smeets (2013) [2]. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in tt.

KW - Multidimensional continued fraction

KW - Minkowski reduction

KW - LLL-reduction

U2 - 10.1016/j.indag.2014.04.003

DO - 10.1016/j.indag.2014.04.003

M3 - Article

SN - 0019-3577

VL - 25

SP - 632

EP - 645

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

IS - 4

ER -

Geodesic continued fractions and LLL

Abstract

Keywords

Access to Document

Fingerprint

Cite this