The Borel—Bernstein Theorem for multidimensional continued fractions

2001 ◽  
Vol 85 (1) ◽  
pp. 1-41 ◽  
Author(s):  
Arnaldo Nogueira
Keyword(s):  
Continued Fractions ◽  
Bernstein Theorem ◽  
Multidimensional Continued Fractions

scholarly journals A generalized family of multidimensional continued fractions: TRIP Maps

2014 ◽  
Vol 10 (08) ◽  
pp. 2151-2186 ◽  
Author(s):  
Krishna Dasaratha ◽  
Laure Flapan ◽  
Thomas Garrity ◽  
Chansoo Lee ◽  
Cornelia Mihaila ◽  
...  
Keyword(s):  
Number Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Real Numbers ◽  
Multidimensional Continued Fractions ◽  
The Family ◽  
Before And After ◽  
Multidimensional Continued Fraction

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.

A 2-DIMENSIONAL ALGORITHM RELATED TO THE FAREY–BROCOT SEQUENCE

2012 ◽  
Vol 08 (01) ◽  
pp. 149-160
Author(s):  
F. SCHWEIGER
Keyword(s):  
Continued Fractions ◽  
Dual Algorithm ◽  
Ergodic Properties ◽  
Minkowski Function ◽  
Special Cases ◽  
Multidimensional Continued Fractions ◽  
New Type ◽  
Interesting Generalization

Moshchevitin and Vielhaber gave an interesting generalization of the Farey–Brocot sequence for dimension d ≥ 2 (see [N. Moshchevitin and M. Vielhaber, Moments for generalized Farey–Brocot partitions, Funct. Approx. Comment. Math.38 (2008), part 2, 131–157]). For dimension d = 2 they investigate two special cases called algorithm [Formula: see text] and algorithm [Formula: see text]. Algorithm [Formula: see text] is related to a proposal of Mönkemeyer and to Selmer algorithm (see [G. Panti, Multidimensional continued fractions and a Minkowski function, Monatsh. Math.154 (2008) 247–264]). However, algorithm [Formula: see text] seems to be related to a new type of 2-dimensional continued fractions. The content of this paper is first to describe such an algorithm and to give some of its ergodic properties. In the second part the dual algorithm is considered which behaves similar to the Parry–Daniels map.

scholarly journals On $p$-adic multidimensional continued fractions

2019 ◽  
Vol 88 (320) ◽  
pp. 2913-2934
Author(s):  
Nadir Murru ◽  
Lea Terracini
Keyword(s):  
Continued Fractions ◽  
Multidimensional Continued Fractions
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