Multidimensional Continued Fraction Research Articles

Generating New Partition Identities via a Generalized Continued Fraction Algorithm

Using the slow triangle map (a type of multi-dimensional continued fraction algorithm), we exhibit a method for generating any number of new identities for subsets of integer partitions.

Combinatorial properties of multidimensional continued fractions

The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization arising from an algorithm due to Jacobi, have been poorly investigated in this sense, up to now. In this paper, we propose a combinatorial interpretation of the convergents of multidimensional continued fractions in terms of counting some particular tilings, generalizing some results that hold for classical continued fractions.

Invariant measures in non-conformal fibered systems with singularities

We study a large class of transformations FT which are only piecewise differentiable on countably many pieces and also non-conformal. There exist random countably generated limit sets JT,ω in the fibers of FT. We prove a Global Volume Lemma implying that the push-forward measures of equilibrium measures from a countable shift space, are exact dimensional on the non-compact global basic set JT. A dimension formula of Ledrappier-Young type is obtained for these measures, by using their Lyapunov exponents and marginal entropies. Then, we study the geometric potentials ψT,s on ΣI, and we prove that the dimensions of the push-forward measures νsω in fibers are independent of ω, and they depend real-analytically on the parameter s from an interval F(T). Moreover, we establish a Variational Principle for dimension in fibers. Our results apply in particular to new types of multidimensional continued fractions.

Multidimensional continued fractions and symbolic codings of toral translations

It has been a long-standing problem to find good symbolic codings for translations on the d -dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy’s approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples include classical algorithms like the Jacobi–Perron, Brun, Cassaigne–Selmer, and Arnoux–Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the 2-dimensional torus having factor complexity 2n + 1 that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all 3-dimensional toral translations having multiscale bounded remainder sets.

Almost everywhere balanced sequences of complexity 2n + 1

We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set $\{1,2\}^\mathbb{N}$ of directive sequences. For a given set $\mathcal{C}$ of two substitutions, we show that there exists a $\mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $\mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $\mu$-almost every $\mathcal{C}$-adic sequence is balanced, where $\mu$ is any shift-invariant ergodic Borel probability measure on $\{1,2\}^\mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $\mu$ is negative.

On Gauss--Kuzmin statistics and the transfer operator for a multidimensional continued fraction algorithm: the triangle map

On Gauss--Kuzmin statistics and the transfer operator for a multidimensional continued fraction algorithm: the triangle map

Proper cyclic symmetries of multidimensional continued fractions

We show that palindromic continued fractions exist in an arbitrary dimension. For dimension $n=4$ we also prove a criterion for an algebraic continued fraction to have a proper cyclic palindromic symmetry. Klein polyhedra are considered as multidimensional generalizations of continued fractions. Bibliography: 11 titles.

Proper cyclic symmetries of multidimensional continued fractions

Работа посвящена доказательству утверждения о существовании в произвольной размерности палиндромичных цепных дробей. Кроме того, доказывается критерий наличия у алгебраической цепной дроби собственной циклической палиндромической симметрии в случае $n=4$. В качестве многомерного обобщения цепных дробей рассматриваются полиэдры Клейна. Библиография: 11 названий.

Asymmetric circular graph with Hosoya index and negative continued fractions

It has been known that the Hosoya index of caterpillar graph can be calculated as the numerator of the simple continued fraction. Recently in [MATCH Commun. Math. Comput. Chem. 2020, 84 (2), 399-428], the author introduces a more general graph called caterpillar-bond graph and shows that its Hosoya index can be calculated as the numerator of the general continued fraction.
 In this paper, we show how the Hosoya index of the graph with non-uniform ring structure can be calculated from the negative continued fraction. We also give the relation between some radial graphs and multidimensional continued fractions in the sense of the Hosoya index.

Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions

Unlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of p-adic numbers mathbb Q_p. Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in mathbb R by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in mathbb Q_p. We focus on the dimension two and study the quality of the simultaneous approximation to two p-adic numbers provided by p-adic MCFs, where p is an odd prime. Moreover, given algebraically dependent p-adic numbers, we see when infinitely many simultaneous approximations satisfy the same algebraic relation. This also allows to give a condition that ensures the finiteness of the p-adic Jacobi–Perron algorithm when it processes some kinds of mathbb Q-linearly dependent inputs.

Ergodicity for [formula omitted]-adic continued fraction algorithms

Following Schweiger’s generalization of multidimensional continued fraction algorithms, we consider a very large family of p-adic multidimensional continued fraction algorithms, which include Schneider’s algorithm, Ruban’s algorithms, and the p-adic Jacobi–Perron algorithm as special cases. The main result is to show that all the transformations in the family are ergodic with respect to the Haar measure.

Functional analysis behind a family of multidimensional continued fractions. Part II

This paper is a direct continuation of Functional analysis behind a Family of Multidimensional Continued Fractions: Part I, in which we started the exploration of the functional analysis behind the transfer operators for triangle partition maps, a family that includes many, if not most, well-known multidimensional continued fraction algorithms. This allows us now to find eigenfunctions of eigenvalue 1 for transfer operators associated with select triangle partition maps on specified Banach spaces. We proceed to prove that the transfer operators, viewed as acting on one-dimensional families of Hilbert spaces, associated with select triangle partition maps are nuclear of trace class zero. We finish by deriving Gauss-Kuzmin distributions associated with select triangle partition maps.

Simplicity of spectra for certain multidimensional continued fraction algorithms

We introduce a new strategy to prove simplicity of the spectrum of Lyapunov exponents that can be applied to a wide class of Markovian multidimensional continued fraction algorithms. As an application we use it for Selmer algorithm in dimension 2 and for the Triangle sequence algorithm and show that these algorithms are not optimal.

Functional analysis behind a family of multidimensional continued fractions. Part I

This paper is a direct continuation of "Functional analysis behind a Family of Multidimensional Continued Fractions: Part I," in which we started the exploration of the functional analysis behind the transfer operators for triangle partition maps, a family that includes many, if not most, well-known multidimensional continued fraction algorithms. This allows us now to find eigenfunctions of eigenvalue 1 for transfer operators associated with select triangle partition maps on specified Banach spaces. We proceed to prove that the transfer operators, viewed as acting on one-dimensional families of Hilbert spaces, associated with select triangle partition maps are nuclear of trace class zero. We finish by deriving Gauss-Kuzmin distributions associated with select triangle partition maps.

On the second Lyapunov exponent of some multidimensional continued fraction algorithms

We study the strong convergence of certain multidimensional continued fraction algorithms. In particular, in the two-dimensional case, we prove that the second Lyapunov exponent of Selmer's algorithm is negative and bound it away from zero. Moreover, we give heuristic results on several other continued fraction algorithms. Our results indicate that all classical multidimensional continued fraction algorithms cease to be strongly convergent for high dimensions. The only exception seems to be the Arnoux-Rauzy algorithm which, however, is defined only on a set of measure zero.

Symmetries of a two-dimensional continued fraction

We describe the symmetry group of a multidimensional continued fraction. As a multidimensional generalization of continued fractions we consider Klein polyhedra. We distinguish two types of symmetries: Dirichlet symmetries, which correspond to the multiplication by units of the respective extension of , and so-called palindromic symmetries. The main result is a criterion for a two-dimensional continued fraction to have palindromic symmetries, which is analogous to the well-known criterion for the continued fraction of a quadratic irrationality to have a symmetric period.

On the finiteness and periodicity of the 𝑝-adic Jacobi–Perron algorithm

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron to obtain periodic representations for algebraic irrationals, analogous to the case of simple continued fractions and quadratic irrationals. Continued fractions have been studied in the field ofpp-adic numbersQp\mathbb {Q}_p. MCFs have also been recently introduced inQp\mathbb {Q}_p, including, in particular, app-adic Jacobi–Perron algorithm. In this paper, we address two of the main features of this algorithm, namely its finiteness and periodicity. Regarding the finiteness of thepp-adic Jacobi–Perron algorithm, our results are obtained by exploiting properties of some auxiliary integer sequences. It is known that a finitepp-adic MCF representsQ\mathbb Q-linearly dependent numbers. However, we see that the converse is not always true and we prove that in this case infinitely many partial quotients of the MCF havepp-adic valuations equal to−1-1. Finally, we show that a periodic MCF of dimensionmmconverges to an algebraic irrational of degree less than or equal tom+1m+1; for the casem=2m=2, we are able to give some more detailed results.

Klein Sail and Diophantine Approximation of a Vector

In the papers by V. I. Arnold and his successors based upon the ideas of H. Poincare and F. Klein, it was the Klein sail associated with an operator in ℝn that they considered to play the role of a multidimensional continued fraction, and in these terms generalizations of Lagrange’s theorem on continued fractions were formulated. A different approach to generalization of the notion of continued fraction was based upon modifications of Euclid’s algorithm for constructing, given an irrational vector, an approximating sequence of rational vectors. We suggest a modification of the Klein sail that is constructed directly from a vector, without any operator. We introduce a numeric characteristic of a Klein sail, its asymptotic anisotropy associated with a one-parameter transformation semigroup of the lattice generating the sail and of its Voronoi cell. In terms of this anisotropy, we hope to give a geometric characterization of irrational vectors worst approximated by rational ones. In the three-dimensional space, we suggest a vector (related to the smallest Pisot–Vijayaraghavan number) that is a candidate for this role. This vector may be called an analog of the golden number, which is the real number worst approximated by rationals in the classical theory of Diophantine approximations.

Multidimensional continued fractions for cyclic quotient singularities and Dedekind sums

We present a multidimensional continued fraction of Hirzebruch–Jung type which controls a certain resolution of an isolated cyclic quotient singularity, and we study its geometric properties. As an application, we explicitly express a certain 3-dimensional Fourier–Dedekind sum in terms of our continued fraction.

Unimodular Invariance of Karyon Expansions of Algebraic Numbers in Multidimensional Continued Fractions

By the method of differentiation of induced toric tilings, periodic expansions for algebraic irrationalities in multidimensional continued fractions are found. These expansions give the best karyon approximations with respect to polyhedral norms. The above irrationalities are obtained by the composition of backward continued fraction mappings and unimodular transformations of algebraic units that are expanded in purely periodic continued fractions. Karyon expansions have several invariants: recurrence relations for the numerators and denominators of the convergents of continued fractions and the rate of multidimensional approximation of irrationalities by rational numbers.