A generalized family of multidimensional continued fractions: TRIP Maps

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

doi:10.1142/s1793042114500730

Cited by 13 publications

(41 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof is almost exactly the same as the proof in Theorem 6.2 in [6]. That proof shows that there is an invertible 3 × 3 matrix A with integer entries so that…”

Section: Barycentric Triangle Partition Mapsmentioning

confidence: 68%

“…Thus, Panti turns to the ndimensional generalizations of these maps, known as the Mönkemeyer map (M) and the tent map (T ) to find his generalization of ?(x). (At this point the reader should note that Dasratha et al showed in [6] that the Mönkemeyer map in the 2-dimensional case corresponds to the TRIP map T (e,132,23) , a fact that we will later be exploiting.) Defining △ as the n-dimensional simplex that reduces to our familiar triangle in 2-dimensions, he presents the following…”

Section: Panti's Workmentioning

confidence: 84%

“…The above development of the triangle sequence is called the additive approach. In [9,5,6,7,14], the multiplicative version is used. In the multiplicative version, we look at the subtriangles…”

Section: Additive Versus Multiplicativementioning

confidence: 99%

“…We now reduce the above 21 cases to just 15. We present the following lemma from [6] Proof. The permutation (e, 23, 23) gives matrices:…”

Section: Reduction To 15 Casesmentioning

confidence: 99%

“…Recently many if not most known multidimensional continued fraction algorithms have been put into a common framework of a family of algorithms, which are called triangle partition maps (TRIP maps) [6]. It is natural to consider if there are analogs of the Minkowksi question mark function for each TRIP map, and indeed this is the goal of this paper.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Generalizing the Minkowski question mark function to a family of multidimensional continued fractions

Garrity

McDonald

2018

Int. J. Number Theory

Self Cite

View full text Add to dashboard Cite

The Minkowski question mark function ? : [0, 1] → [0, 1] is a continuous, strictly increasing, one-to-one and onto function that has derivative zero almost everywhere. Key to these facts are the basic properties of continued fractions. Thus ?(x) is a naturally occurring number theoretic singular function. This paper generalizes the question mark function to the 216 triangle partition (TRIP) maps. These are multidimensional continued fractions which generate a family of almost all known multidimensional continued fractions. We show for each TRIP map that there is a natural candidate for its analog of the Minkowski question mark function. We then show that the analog is singular for 96 of the TRIP maps and show that 60 more are singular under an assumption of ergodicity.

show abstract

“…The proof is almost exactly the same as the proof in Theorem 6.2 in [6]. That proof shows that there is an invertible 3 × 3 matrix A with integer entries so that…”

Section: Barycentric Triangle Partition Mapsmentioning

confidence: 68%

Section: Panti's Workmentioning

confidence: 84%

Section: Additive Versus Multiplicativementioning

confidence: 99%

“…We now reduce the above 21 cases to just 15. We present the following lemma from [6] Proof. The permutation (e, 23, 23) gives matrices:…”

Section: Reduction To 15 Casesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generalizing the Minkowski question mark function to a family of multidimensional continued fractions

Garrity

McDonald

2018

Int. J. Number Theory

Self Cite

View full text Add to dashboard Cite

show abstract

Cubic irrationals and periodicity via a family of multi-dimensional continued fraction algorithms

et al. 2014

Self Cite

View full text Add to dashboard Cite

We construct a countable family of multi-dimensional continued fraction algorithms, built out of five specific multidimensional continued fractions, and find a wide class of cubic irrational real numbers α so that either (α, α 2 ) or (α, α − α 2 ) is purely periodic with respect to an element in the family. These cubic irrationals seem to be quite natural, as we show that, for every cubic number field, there exists a pair (u, u ′ ) with u a unit in the cubic number field (or possibly the quadratic extension of the cubic number field by the square root of the discriminant) such that (u, u ′ ) has a periodic multidimensional continued fraction expansion under one of the maps in the family generated by the initial five maps. These results are built on a careful technical analysis of certain units in cubic number fields and our family of multi-dimensional continued fractions. We then recast the linking of cubic irrationals with periodicity to the linking of cubic irrationals with the construction of a matrix with nonnegative integer entries for which at least one row is eventually periodic.

show abstract

Linear recurrence sequences and periodicity of multidimensional continued fractions

Murru

2016

Ramanujan J

View full text Add to dashboard Cite

Multidimensional continued fractions generalize classical continued fractions with the aim of providing periodic representations of algebraic irrationalities by means of integer sequences. However, there does not exist any algorithm that provides a periodic multidimensional continued fraction when algebraic irrationalities are given as inputs. In this paper, we provide a characterization for periodicity of Jacobi-Perron algorithm by means of linear recurrence sequences. In particular, we prove that partial quotients of a multidimensional continued fraction are periodic if and only if numerators and denominators of convergents are linear recurrence sequences, generalizing similar results that hold for classical continued fractions.

show abstract

A generalized family of multidimensional continued fractions: TRIP Maps

Cited by 13 publications

References 7 publications

Generalizing the Minkowski question mark function to a family of multidimensional continued fractions

Generalizing the Minkowski question mark function to a family of multidimensional continued fractions

Cubic irrationals and periodicity via a family of multi-dimensional continued fraction algorithms

Linear recurrence sequences and periodicity of multidimensional continued fractions

Contact Info

Product

Resources

About