相关论文: A Geometric Interpretation of the $p$-adic Littlew…
In this paper we adopt a geometric point of view regarding a famous conjecture due to Littlewood in diophantine approximation of real numbers. Following the spirit of the geometric theory of continued fractions, we give a sufficient…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
Continued fractions have a long history in number theory, especially in the area of Diophantine approximation. The aim of this expository paper is to survey the main results on the theory of $p$--adic continued fractions, i.e. continued…
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…
Continued fractions have been long studied due to their strong properties, such as rational approximation. In this extent, their arithmetic over real numbers has represented an intriguing problem throughout the years. In this paper, we…
The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…
We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give,…
Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question…
In a recent paper A.Beardon and I.Short proposed to use chains of tangent horocycles as an extended tool describing continued fractions. We review the origin of such construction from the Moebius transformations point of view. Related…
We introduce here a general framework for studying continued fraction expansions for complex numbers and establish some results on the convergence of the corresponding sequence of convergents. For continued fraction expansions with partial…
We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend, and use it…
In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of…
The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real…
Most well-known multidimensional continued fractions, including the M\"{o}nkemeyer map and the triangle map, are generated by repeatedly subdividing triangles. This paper constructs a family of multidimensional continued fractions by…
We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…
Continued fractions in the field of $p$--adic numbers have been recently studied by several authors. It is known that the real continued fraction of a positive quadratic irrational is eventually periodic (Lagrange's Theorem). It is still…
In this paper, we apply results on number systems based on continued fraction expansions to modular arithmetic. We provide two new algorithms in order to compute modular multiplication and modular division. The presented algorithms are…
We develop a theory of $p$-adic continued fractions for a quaternion algebra $B$ over $\mathbb Q$ ramified at a rational prime $p$. Many properties holding in the commutative case can be proven also in this setting. In particular, we focus…
Consider the representation of a rational number as a continued fraction, associated with "odd" Euclidean algorithm. In this paper we prove certain properties for the limit distribution function for sequences of rationals with bounded sum…
$p$-adic continued fractions, as an extension of the classical concept of classical continued fractions to the realm of $p$-adic numbers, offering a novel perspective on number representation and approximation. While numerous $p$-adic…