Related papers: Duke's Theorem and Continued Fractions
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…
We establish a coboundary condition for a sequence of ergodic sums (i.e.~Birkhoff partial sums) to be almost surely uniformly distributed mod $1$. Applications are given when the sequence is generated by a Gibbs-Markov map. In particular,…
Let $\wh K$ be the field of formal Laurent series in $X^{-1}$ over the finite field $k$, and let $A$ be the ring of polynomials in $X$ over $k$. One of the main results of the paper is to give a particularly nice coding of the geodesic flow…
The continued fraction mapping maps a number in the interval $[0,1)$ to the sequence of its partial quotients. When restricted to the set of irrationals, which is a subspace of the Euclidean space $\mathbb{R}$, the continued fraction…
To compare continued fraction digits with the denominators of the corresponding approximants we introduce the arithmetic-geometric scaling. We will completely determine its multifractal spectrum by means of a number theoretical free energy…
We determine the classical approximation constants $w_{3}(\zeta),w_{3}^{\ast}(\zeta),\lambda_{3}(\zeta)$ such as the uniform constants $\widehat{w}_{3}(\zeta),\widehat{w}_{3}^{\ast}(\zeta),\widehat{\lambda}_{3}(\zeta)$ associated to real…
We show that the growth rate of denominator $Q_n$ of the $n$-th convergent of negative expansion of $x$ and the rate of approximation: $$ \frac{\log{n}}{n}\log{\left|x-\frac{P_n}{Q_n}\right|}\rightarrow -\frac{\pi^2}{3} \quad \text{in…
This paper aims to introduce high school students to the intriguing world of continued fractions, a mathematical concept that provides a unique representation of numbers. The study focuses on the exploration and development of the…
For any real number $x \in [0,1)$, we denote by $q_n(x)$ the denominator of the $n$-th convergent of the continued fraction expansion of $x$ $(n \in \mathbb{N})$. It is well-known that the Lebesgue measure of the set of points $x \in [0,1)$…
We prove the convergence of a wide class of continued fractions, including generalized continued fractions over quaternions and octonions. Fractional points in these systems are not bounded away from the unit sphere, so that the iteration…
This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the…
We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P^1R, determines a…
We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 + >...))) where the s_n are independent random variables with the same gamma distribution. For every realisation of the sequence, S(t) defines a Stieltjes function.…
We prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by F. Schweiger and studied also by C. Kraaikamp and A.…
We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To…
This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an…
We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the…
We introduce the concept of Minkowski normality, a different type of normality for the regular continued fraction expansion. We use the ordering \[ \frac{1}{2},\quad \frac{1}{3}, \frac{2}{3},\quad \frac{1}{4}, \frac{3}{4},\frac{2}{5},…
Generalizing a result of Pourchet, we prove that, if $\alpha,\beta$ are power sums satisfying suitable conditions, the length of the continued fraction of the ratio $\alpha(n)/\beta(n)$ tends to infinity with $n$.
We show that for a fixed integer $n \neq \pm2$, the congruence $x^2 + nx \pm 1 \equiv 0 \pmod{\alpha}$ has the solution $\beta$ with $0 < \beta < \alpha$ if and only if $\alpha/\beta$ has a continued fraction expansion with sequence of…