TY - JOUR
T1 - A generalized family of multidimensional continued fractions
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 - Funding Information:
The authors thank the National Science Foundation for their support of this research via Grant DMS-0850577. We would like to thank Ilya Amburg for catching some mistakes and would like to thank the referee for helpful comments.
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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U2 - 10.1142/S1793042114500730
DO - 10.1142/S1793042114500730
M3 - Article
AN - SCOPUS:84928290288
SN - 1793-0421
VL - 10
SP - 2151
EP - 2186
JO - International Journal of Number Theory
JF - International Journal of Number Theory
IS - 8
ER -