Related papers: Complex continued fractions, Kleinian and extremal…
We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aimed to compare the efficiency by describing the rate at which the…
We establish a central limit theorem for counting large continued fraction digits $(a_n)$, i.e. we count occurrences $\{a_n>b_n\}$, where $(b_n)$ is a sequence of positive integers. Our result improves a similar result by Philipp which…
In this paper we study the properties of an algorithm for generating continued fractions in the field of p-adic numbers $\mathbb{Q}_p$. First of all, we obtain an analogue of the Galois' Theorem for classical continued fractions. Then, we…
We study the bit complexity of two methods, related to the Euclidean algorithm, for computing cubic and quartic analogs of the Jacobi symbol. The main bottleneck in such procedures is computation of a quotient for long division. We give…
Classical results on Diophantine approximation, such as Roth's theorem, provide the most effective techniques for proving the transcendence of special kinds of continued fractions. Multidimensional continued fractions are a generalization…
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…
Quadratic irrationals posses a periodic continued fraction expansion. Much less is known about cubic irrationals. We do not even know if the partial quotients are bounded, even though extensive computations suggest they might follow…
Adolf Hurwitz proposed in 1887 a continued fraction algorithm for complex numbers: Hurwitz continued fractions (HCF). Among other similarities between HCF and regular continued fractions, quadratic irrational numbers over $\mathbb{Q}(i)$…
This paper continues the author's previous studies on continued fractions and Heron's algorithm, as from his former JMM2017 presentation (see \cite{CF.HA}).\par\medskip Extending the notion of continued fraction to the $p$-adic fields, one…
In 2008, Dan Romik studied in this journal Primitive Pythagorean Triples, or PPTs. In order to do so, he introduced a modified slow (subtractive) Euclidean algorithm, and showed that the underlying dynamical system of this Euclidean…
Computing well-rounded twists of ideals in number fields has been done when the field degree is $2$. In this paper, we develop a new algorithm to detect whether a basis of an ideal $\mathfrak{I}$ in a cyclic cubic field $F$ yields a…
We study, from the viewpoint of metrical number theory and (infinite) ergodic theory, the probabilistic laws governing the occurrence of prime numbers as digits in continued fraction expansions of real numbers.
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 give continued fraction expansions of the generating functions of Bernoulli numbers, Cauchy numbers, Euler numbers, harmonic numbers, and their generalized or related numbers. In particular, we focus on explicit forms of the convergents…
This paper concerns extension of the classical Lagrange theorem, on the eventual periodicity of continued fraction expansions of quadratic surds, and the versions of it found in the literature in the case of complex numbers. In this…
We develop algorithms to turn quotients of rings of rings of integers into effective Euclidean rings by giving polynomial algorithms for all fundamental ring operations. In addition, we study normal forms for modules over such rings 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…
We address the problem of determining the Hausdorff dimension of sets consisting of complex irrationals whose complex continued fraction digits satisfy prescribed restrictions and growth conditions. For the Hurwitz continued fraction, we…
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…
This paper is a sequel to our previous work in which we found a combinatorial realization of continued fractions as quotients of the number of perfect matchings of snake graphs. We show how this realization reflects the convergents of the…