Related papers: Classical homogeneous multidimensional continued f…
A basic result in the elementary theory of continued fractions says that two real numbers share the same tail in their continued fraction expansions iff they belong to the same orbit under the projective action of PGL(2,Z). This result was…
We introduce the notion of matrices graph, defining continued fraction algorithms where the past and the future are almost independent. We provide an algorithm to convert more general algorithms into matrices graphs. We present an algorithm…
We prove a suite of dynamical results, including exactness of the transformation and piecewise-analyticity of the invariant measure, for a family of continued fraction systems, including specific examples over reals, complex numbers,…
In this paper, we present some generalizations of Lagrange's theorem in the classical theory of continued fractions motivated by the geometric interpretation of the classical theory in terms of closed geodesics on the modular curve. As a…
A Lagrange Theorem in dimension 2 is proved, for a particular two-dimensional algorithm, with a very natural geometrical definition. Dirichlet-type properties for the convergence of the algorithm are also proved. These properties procced…
For a given sequence of positive integers we make an explicit construction of a reduced hyperbolic operator in SL(2,z) with the sequence as a period of a geometric continued fraction in the sense of Klein. Further we experimentally study an…
This paper present the important role that random system with complete connections played in solving the Gauss problem associated to the regular continued fractions. Hence, using the ergodic behavior of homogeneous random system with…
Suppose we partition the integers into finitely many cells. Can we always find a solution of the equation $x^2+y^2=z^2$ with $x,y,z$ on the same cell? What about more general homogeneous quadratic equations in three variables? These are…
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…
In this article we study a renormalization scheme with which we find all semi-regular continued fractions of a number in a natural way. We define two maps, T_slow and T_fast: these maps are defined for (x,y) in [0,1], where x is the number…
This work addresses the problem of simulating Gaussian random fields that are continuously indexed over a class of metric graphs, termed graphs with Euclidean edges, being more general and flexible than linear networks. We introduce three…
Proper continued fractions are generalized continued fractions with positive integer numerators $a_i$ and integer denominators with $b_i\geq a_i$. In this paper we study the strength of approximation of irrational numbers to their…
We develop a continued fraction algorithm in finite extensions of $\Q_p$ generalising certain algorithms in $\Q_p$, and prove the finiteness property for certain small degree extensions. We also discuss the metrical properties of the…
This study provides new results about the probabilistic behaviour of a class of Euclidean algorithms: the asymptotic distribution of a whole class of cost-parameters associated to these algorithms is normal. For the cost corresponding to…
The distributional analysis of Euclidean algorithms was carried out by Baladi and Vall\'{e}e. They showed the asymptotic normality of the number of division steps and associated costs in the Euclidean algorithm as a random variable on the…
We statistically compare the relationships between frequencies of digits in continued fraction expansions of typical rational points in the unit interval and higher dimensional generalisations. This takes the form of a Large Deviation and…
We introduce and study in detail a special class of backward continued fractions that represents a generalization of R\'enyi continued fractions. We investigate the main metrical properties of the digits occurring in these expansions and we…
This paper discusses the irreducibility and geometric ergodicity of the Hamiltonian Monte Carlo (HMC) algorithm. We consider cases where the number of steps of the symplectic integrator is either fixed or random. Under mild conditions on…
A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued…
We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves…