Simple continued fractions and cutting sequences

Abstract

The connection between cutting sequences of a directed geodesic in the tessellated hyperbolic plane ?², the modular group Γ = PSL(2, ?) and the simple continued fractions of an end point w of the geodesic have been established by Series [13]. In this paper we represent the simple continued fractions of w ∈ ? and the “L” and “R” codes of the cutting sequence in terms of modular and extended modular transformations. We will define a T ₀-path on a graph whose vertices are the set of Farey triangles, as the equivalent of the cutting sequence. The relationship between the directed geodesic with end point w on ?, the Farey tessellation and the simple continued fraction expansion of w ∈ ? _∞ then follows easily as a consequence of this redefinition. Finite, infinite and periodic simple continued fractions are subsequently examined in this light.     

Continued fractions for hyperquadratic power series over a finite field

An irrational power series over a finite field of characteristic p is called hyperquadratic if it satisfies an algebraic equation of the form x=(Ax^r+B)/(Cx^r+D), where r is a power of p and the coefficients belong to . These algebraic power series are analogues of quadratic real numbers. This analogy makes their continued fraction expansions specific as in the classical case, but more sophisticated. Here we present a general result on the way some of these expansions are generated. We apply it to describe several families of expansions having a regular pattern.     

Sharp Bounds for Symmetric and Asymmetric Diophantine Approximation

In 2004, Tong found bounds for the approximation quality of a regular continued fraction convergent to a rational number, expressed in bounds for both the previous and next approximation. The authors sharpen his results with a geometric method and give both sharp upper and lower bounds. The asymptotic frequencies that these bounds occur are also calculated.     

Continued fractions with bounded partial quotients

This paper gives the exact bound of the continued fraction expansion of when has bounded partial quotients and is a Möbius transformation where all entries are integers.

     

Continued fractions with three limit points

On page 45 in his lost notebook, Ramanujan asserts that a certain q-continued fraction has three limit points. More precisely, if A_n/B_n denotes its nth partial quotient, and n tends to ∞ in each of three residue classes modulo 3, then each of the three limits of A_n/B_n exists and is explicitly given by Ramanujan. Ramanujan's assertion is proved in this paper. Moreover, general classes of continued fractions with three limit points are established.     

The convergence behavior of q-continued fractions on the unit circle

In a previous paper, we showed the existence of an uncountable set of points on the unit circle at which the Rogers-Ramanujan continued fraction does not converge to a finite value. In this present paper, we generalise this result to a wider class of q-continued fractions, a class which includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fractions. We show, for each q-continued fraction, G(q), in this class, that there is an uncountable set of points, Y _G , on the unit circle such that if y ? Y _G then G(y) does not converge to a finite value. We discuss the implications of our theorems for the convergence of other q-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.     

A thermodynamic classification of real numbers

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A new classification scheme for real numbers is given, motivated by ideas from statistical mechanics in general and work of Knauf (1993) [16] and Fiala and Kleban (2005) [8] in particular. Critical for this classification of a real number will be the Diophantine properties of its continued fraction expansion.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=qnPF2QS4cRg.     

Continued fraction expansions of matrix eigenvectors

We examine various properties of the continued fraction expansions of matrix eigenvector slopes of matrices from the SL(2, ℤ) group. We calculate the average period length, maximum period length, average period sum, maximum period sum, and the distributions of 1s, 2s, and 3s in the periods versus the radius of the ball within which the matrices are located. We also prove that the periods of continued fraction expansions from the real irrational roots of x ²+px+q=0 are always palindromes.      

Some remarks on a probability limit theorem for continued fractions

It is shown that if a certain condition on the variances of the partial sums is satisfied then a theorem of Philipp and Stout, which implies the asymptotic fluctuation results known for independent random variables, can be applied to some quantities related to continued fractions. Previous results on the behavior of the approximation by the continued fraction convergents to a random real number are improved.

     

Explicit continued fractions with expected partial quotient growth

For let be the continued fraction expansion of . Write

We construct some numbers 's with

     

An algorithm of infinite sums representations and Tasoev continued fractions

For any given real number, its corresponding continued fraction is unique. However, given an arbitrary continued fraction, there has been no general way to identify its corresponding real number. In this paper we shall show a general algorithm from continued fractions to real numbers via infinite sums representations. Using this algorithm, we obtain some new Tasoev continued fractions.

     

Rates of convergence for continued fractions of irrational numbers

The dynamics of the Gauss Map suggests a way to compare the convergence to a real number ζ ε(0,l) of a continued fraction and the divergence of the orbit of ζ Of particular interest is the comparison of the rate of convergence to ζ of its simple continued fraction and the rate of divergence by the Gauss Map of the orbit of ζ for all irrational numbers in (0,l). We state and prove sharp inequalities for the convergence of the sequence of rational convergents of an irrational number ζ. We show that the product of the rate of convergence of the continued fraction of ζ and the rate of divergence by the Gauss Map of the orbit of ζ equals 1.     

On distinct unit fractions whose sum equals 1

Questions, partial and complete answers about the diophantine equation in distinct positive integers are given when additional requirements are asked on the x_i's such as: being large, odd, even or x_ix_j for i≠j. Various combinations of the above conditions are also considered.     

A note on bounds for the derivatives of continued fractions

Although it is difficult to differentiate analytic functions defined by continued fractions, it is relatively easy in some cases to determine uniform bounds on such derivatives by perceiving the continued fraction as an infinite composition of linear fractional transformations and applying an infinite chain rule for differentiation.     

Continued fractions in local fields, II

The present paper is a continuation of an earlier work by the author. We propose some new definitions of -adic continued fractions. At the end of the paper we give numerical examples illustrating these definitions. It turns out that for every if then has a periodic continued fraction expansion. The same is not true in for some larger values of

     

Bounding differences in Jager Pairs

Symmetrical subdivisions in the space of Jager Pairs for continued fractions-like expansions will provide us with bounds on their differences. Results will also apply to the classical regular and backwards continued fractions expansions, which are realized as special cases.     

Metrical diophantine approximation for continued fraction like maps of the interval

We study the metrical properties of a class of continued fraction-like mappings of the unit interval, each of which is defined as the fractional part of a Möbius transformation taking the endpoints of the interval to zero and infinity.

     

Continued fractions with multiple limits

For integers m2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern–Stolz theorem.We give a theorem on a class of Poincaré-type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity.We also generalize a curious q-continued fraction of Ramanujan's with three limits to a continued fraction with k distinct limit points, k2. The k limits are evaluated in terms of ratios of certain q-series.Finally, we show how to use Daniel Bernoulli's continued fraction in an elementary way to create analytic continued fractions with m limit points, for any positive integer m2.