Related papers: The random continued fraction transformation
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…
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization…
We present and develop different approaches to study the asymptotic behavior of the distribution functions in the odd continued fractions case. Firstly, by considering the transition operator of the Markov chain associated with these…
In this work, we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin-Iosifescu…
Several conjectural continued fractions found with the help of various algorithms are published in this paper.
We study large deviation asymptotics for processes defined in terms of continued fraction digits. We use the continued fraction digit sum process to define a stopping time and derive a joint large deviation asymptotic for the upper and…
In regular dynamics, discrete maps are model presentations of discrete dynamical systems, and they may approximate continuous dynamical systems. Maps are used to investigate general properties of dynamical systems and to model various…
We present a one-parameter family of continuous, piecewise affine, area preserving maps of the square, which are inspired by a dynamical system in game theory. Interested in the coexistence of stochastic and (quasi-)periodic behaviour, we…
This paper gives a brief introduction to some important fractional and multifractional Gaussian processes commonly used in modelling natural phenomena and man-made systems. The processes include fractional Brownian motion (both standard and…
Continued fractions are linked to Stern's diatomic sequence 0,1,1,2,1,3,2,3,1,4,... (given by the recursion relation a_2n=a_n and a_{2n+1} = a_n + a_{n+1}, where a_0=0 and a_1=1), which has long been known. Using a particular…
We study random dynamical systems on the real line, considering each dynamical system together with the one generated by the inverse maps. We show that there is a duality between forward and inverse behaviour for such systems, splitting…
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…
The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate…
Linear fractional Galton-Watson branching processes in i.i.d.~random environment are, on the quenched level, intimately connected to random difference equations by the evolution of the random parameters of their linear fractional marginals.…
It is well-known that the excursions of a one-dimensional diffusion process can be studied by considering a certain Riccati equation associated with the process. We show that, in many cases of interest, the Riccati equation can be solved in…
We show that the additive-slow-Farey version of the traditional continued fractions algorithm has a natural interpretation as a method for producing integer partitions of a positive number $n$ into two smaller numbers, with multiplicity. We…
Based on the discrete fractional random transform (DFRNT), we present the discrete fractional random cosine and sine transforms (DFRNCT and DFRNST). We demonstrate that the DFRNCT and DFRNST can be regarded as special kinds of DFRNT and…
We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$.…
In this paper, we represent a continued fraction expression of Mathieu series by a continued fraction formula of Ramanujan. As application, we obtain some new bounds for Mathieu series.
We construct an absolutely normal number whose continued fraction expansion is normal in the sense that it contains all finite patterns of partial quotients with the expected asymptotic frequency as given by the Gauss-Kuzmin measure. The…