Related papers: Three examples of three-dimensional continued frac…
This paper is a short survey of the recent results on examples of periodic two-dimensional continued fractions (in Klein's model). In the last part of this paper we formulate some questions, problems and conjectures on geometrical…
In this work we prove a criterion for an algebraic continued fraction to have a proper palindromic symmetry in dimension $4$. As a multidimensional generalization of continued fractions, we consider Klein polyhedra.
In this paper we prove in detail a criterion for an algebraic continued fraction to have a proper palindromic symmetry in dimension $4$. We also present a new proof of the criterion for an algebraic continued fraction to have a proper…
In this paper, we will first summarize known results concerning continued fractions. Then we will limit our consideration to continued fractions of quadratic numbers. The second author described periods and sometimes precise form of…
We consider the geometric generalization of ordinary continued fraction to the multidimensional case introduced by F. Klein in 1895. A multidimensional periodic continued fraction is the union of sails with some special group acting freely…
This work is devoted to the proof of the statement about the existence of palindromic continued fractions in an arbitrary dimension. In addition, it is proved the criterion that an algebraic continued fraction has proper cyclic palindromic…
In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued…
The Lagrange theorem on continued fractions states that a number is a quadratic surd if and only if its continued fraction expansion is eventually periodic. The current paper is devoted to a multidimensional generalization of this fact. As…
Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum…
In this paper we describe the group of symmetries of a two-dimensional continued fraction. As a multidimensional generalization of continued fractions we consider Klein polyhedra. We distinguish two types of symmetries: the Dirichlet-type…
Euler gives a continued fraction representation of (1 + x)n. involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=t sqrt(-1), for "vanishing" n, and for infinite…
Continued fractions whose elements are polynomial sequences have been carefully studied mostly in the cases where the degree of the numerator polynomial is less than or equal to two and the degree of the denominator polynomial is less than…
In this paper, we introduce the polynomial continued fraction, a close relative of the well-known simple continued fraction expansions which are widely used in number theory and in general. While they may not possess all the intriguing…
We give combinatorial descriptions of the terms occurring in continuants of general continued fractions that diverge to three limits. Equating these with the usual combinatorial descriptions due to Euler, Sylvester, and Minding induces…
In this work, we present continued fractions for the arithmetic, geometric, harmonic and cotangent means of $[a_0,a_1,\dots,a_k]$ and $[a_0,a_1,\dots,a_k,a_{k+1}]$, and some of their applications.
Multidimensional Continued Fraction Algorithms are generalizations of the Euclid algorithm and find iteratively the gcd of two or more numbers. They are defined as linear applications on some subcone of $\mathbb{R}^d$. We consider…
For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…
We study continued logarithms as introduced by Bill Gosper and studied by J. Borwein et. al.. After providing an overview of the type I and type II generalizations of binary continued logarithms introduced by Borwein et. al., we focus on a…
We study a natural extension to complex numbers of the standard continued fractions. The basic algorithm is due to Lagrange and Gauss, though it seems to have gone mostly unnoticed as a way to create continued fractions. The new…
Euler gives a continued fraction representation of (1+x)^n involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=t sqrt(-1), for ``vanishing'' n, and for infinite…