A generalized family of multidimensional continued fractions: TRIP Maps

Krishna Dasaratha; Laure Flapan; Thomas Garrity; Chansoo Lee; Cornelia Mihaila; Nicholas Neumann-Chun; Sarah Peluse; Matthew Stoffregen

doi:10.1142/S1793042114500730

A generalized family of multidimensional continued fractions: TRIP Maps

Krishna Dasaratha, Laure Flapan, Thomas Garrity, Chansoo Lee, Cornelia Mihaila, Nicholas Neumann-Chun, Sarah Peluse, Matthew Stoffregen

Mathematics

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

8 Scopus citations

Abstract

Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

Original language	English (US)
Pages (from-to)	2151-2186
Number of pages	36
Journal	International Journal of Number Theory
Volume	10
Issue number	8
DOIs	https://doi.org/10.1142/S1793042114500730
State	Published - Dec 25 2014

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

Keywords

Hermite problem
Jacobi-Perron algorithm
Multidimensional continued fractions
Triangle map

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10.1142/S1793042114500730

Cite this

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title = "A generalized family of multidimensional continued fractions: TRIP Maps",

abstract = "Most well-known multidimensional continued fractions, including the M{\"o}nkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and G{\"u}ting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.",

keywords = "Hermite problem, Jacobi-Perron algorithm, Multidimensional continued fractions, Triangle map",

author = "Krishna Dasaratha and Laure Flapan and Thomas Garrity and Chansoo Lee and Cornelia Mihaila and Nicholas Neumann-Chun and Sarah Peluse and Matthew Stoffregen",

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T2 - TRIP Maps

AU - Dasaratha, Krishna

AU - Flapan, Laure

AU - Garrity, Thomas

AU - Lee, Chansoo

AU - Mihaila, Cornelia

AU - Neumann-Chun, Nicholas

AU - Peluse, Sarah

AU - Stoffregen, Matthew

N1 - Publisher Copyright: © World Scientific Publishing Company.

PY - 2014/12/25

Y1 - 2014/12/25

N2 - Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

AB - Most well-known multidimensional continued fractions, including the Mönkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by permuting the vertices of these triangles before and after each subdivision. We obtain an even larger class of multidimensional continued fractions by composing the maps in the family. These include the algorithms of Brun, Parry-Daniels and Güting. We give criteria for when multidimensional continued fractions associate sequences to unique points, which allows us to determine when periodicity of the corresponding multidimensional continued fraction corresponds to pairs of real numbers being cubic irrationals in the same number field.

KW - Hermite problem

KW - Jacobi-Perron algorithm

KW - Multidimensional continued fractions

KW - Triangle map

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A generalized family of multidimensional continued fractions: TRIP Maps

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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