Continued Fraction Algorithm Research Articles

A renormalization scheme for semi-regular continued fractions

AbstractIn this article we study a renormalization scheme with which we find all semi-regular continued fractions of a number in a natural way. We define two maps, $$\hat{T}_{slow}$$ T ^ slow and $$\hat{T}_{fast}$$ T ^ fast : these maps are defined for $$(x,y)\in [0,1]$$ ( x , y ) ∈ [ 0 , 1 ] , where x is the number for which a semi- regular continued fraction representation is developed by $$\hat{T}_{slow}$$ T ^ slow according to the parameter y. The set of all possible semi-regular continued fraction representations of x are bijectively constructed as the parameter y varies. The map $$\hat{T}_{fast}$$ T ^ fast is a “sped up" version of the map $$\hat{T}_{slow}$$ T ^ slow , and we show that $$\hat{T}_{fast}$$ T ^ fast is ergodic with respect to a probability measure which is mutually absolutely continuous with Lebesgue measure. In contrast, $$\hat{T}_{slow}$$ T ^ slow preserves no such measure, but does preserve an infinite, $$\sigma $$ σ -finite measure mutually absolutely continuous with Lebesgue measure. Furthermore, we generate a sequence of substitutions which generate a symbolic coding of the orbit of y under rotation by x modulo one. In the last section we highlight how our scheme can be used to generate semi-regular continued fractions explicitly for specific continued fraction algorithms such as Nakada’s $$\alpha $$ α -continued fractions.

Purely periodic continued fractions and graph-directed iterated function systems

We describe Gauss-type maps as geometric realizations of certain codes in the monoid of nonnegative matrices in the extended modular group. Each such code, together with an appropriate choice of unimodular intervals in P1R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{P}\\,}}^1\\mathbb {R}$$\\end{document}, determines a dual pair of graph-directed iterated function systems, whose attractors contain intervals and constitute the domains of a dual pair of Gauss-type maps. Our framework covers many continued fraction algorithms (such as Farey fractions, Ceiling, Even and Odd, Nearest Integer, …\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ldots $$\\end{document}) and provides explicit dual algorithms and characterizations of those quadratic irrationals having a purely periodic expansion.

On matching and periodicity for (N,α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(N,\\alpha )$$\\end{document}-expansions

Recently a new class of continued fraction algorithms, the (N,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(N,\\alpha $$\\end{document})-expansions, was introduced in Kraaikamp and Langeveld (J Math Anal Appl 454(1):106–126, 2017) for each N∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\in \\mathbb {N}$$\\end{document}, N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document} and α∈(0,N-1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (0,\\sqrt{N}-1]$$\\end{document}. Each of these continued fraction algorithms has only finitely many possible digits. These (N,α)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(N,\\alpha )$$\\end{document}-expansions ‘behave’ very different from many other (classical) continued fraction algorithms; see also Chen and Kraaikamp (Matching of orbits of certain n-expansions with a finite set of digits (2022). To appear in Tohoku Math. J arXiv:2209.08882), de Jonge and Kraaikamp (Integers 23:17, 2023), de Jonge et al. (Monatsh Math 198(1):79–119, 2022), Nakada (Tokyo J Math 4(2):399–426, 1981) for examples and results. In this paper we will show that when all digits in the digit set are co-prime with N, which occurs in specified intervals of the parameter space, something extraordinary happens. Rational numbers and certain quadratic irrationals will not have a periodic expansion. Furthermore, there are no matching intervals in these regions. This contrasts sharply with the regular continued fraction and more classical parameterised continued fraction algorithms, for which often matching is shown to hold for almost every parameter. On the other hand, for α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} small enough, all rationals have an eventually periodic expansion with period 1. This happens for all α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} when N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=2$$\\end{document}. We also find infinitely many matching intervals for N=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=2$$\\end{document}, as well as rationals that are not contained in any matching interval.

Quantum statistical mechanics and the boundary of modular curves

The theory of limiting modular symbols provides a noncommutative geometric model of the boundary of modular curves that includes irrational points in addition to cusps. A noncommutative space associated to this boundary is constructed, as part of a family of noncommutative spaces associated to different continued fractions algorithms, endowed with the structure of a quantum statistical mechanical system. Two special cases of this family of quantum systems can be interpreted as a boundary of the system associated to the Shimura variety of GL2 and an analog for SL2. The structure of equilibrium states for this family of systems is discussed. In the geometric cases, the ground states evaluated on boundary arithmetic elements are given by pairings of cusp forms and limiting modular symbols.

Generating New Partition Identities via a Generalized Continued Fraction Algorithm

Using the slow triangle map (a type of multi-dimensional continued fraction algorithm), we exhibit a method for generating any number of new identities for subsets of integer partitions.

Regularity properties of k-Brjuno and Wilton functions

AbstractWe study functions related to the classical Brjuno function, namely k-Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We consider various possible versions of them, based on the $$\alpha $$ α -continued fraction developments. We study their BMO regularity properties and their behaviour near rational numbers of their finite truncations. We then complexify the functional equations which they fulfill and we construct analytic extensions of the k-Brjuno and Wilton functions to the upper half-plane. We study their boundary behaviour using an extension of the continued fraction algorithm to the complex plane. We also prove that the harmonic conjugate of the real k-Brjuno function is continuous at all irrational numbers and has a decreasing jump of $$\pi /q^k$$ π / q k at rational points p/q.

Generalizations of Sturmian sequences associated with N-continued fraction algorithms

Given a positive integer N and x∈[0,1]∖Q, an N-continued fraction expansion of x is defined analogously to the classical continued fraction expansion, but with the numerators being all equal to N. Inspired by Sturmian sequences, we introduce the N-continued fraction sequencesω(x,N) and ωˆ(x,N), which are related to the N-continued fraction expansion of x. They are infinite words over a two letter alphabet obtained as the limit of a directive sequence of certain substitutions, hence they are S-adic sequences. When N=1, we are in the case of the classical continued fraction algorithm, and obtain the well-known Sturmian sequences. We show that ω(x,N) and ωˆ(x,N) are C-balanced for some explicit values of C and compute their factor complexity function. We also obtain uniform word frequencies and deduce unique ergodicity of the associated subshifts. Finally, we provide a Farey-like map for N-continued fraction expansions, which provides an additive version of N-continued fractions, for which we prove ergodicity and give the invariant measure explicitly.

On the topological entropy of (a, b)-continued fraction transformations

We study the topological entropy of a two-parameter family of maps related to (a, b)-continued fraction algorithms and prove that it is constant on a square within the parameter space (two vertices of this square correspond to well-studied continued fraction algorithms). The proof uses conjugation to maps of constant slope. We also present experimental evidence that the topological entropy is flexible (i.e. takes any value in a range) on the whole parameter space.

Multidimensional continued fractions and symbolic codings of toral translations

It has been a long-standing problem to find good symbolic codings for translations on the d -dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy’s approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples include classical algorithms like the Jacobi–Perron, Brun, Cassaigne–Selmer, and Arnoux–Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the 2-dimensional torus having factor complexity 2n + 1 that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all 3-dimensional toral translations having multiscale bounded remainder sets.

Decoding Process of RS Codes with Errors and Erasures: An Overview

This paper provides a comprehensive overview of the hard-decision decoding process of Reed-Solomon codes for error-and-erasure decoding. For an [n, k] RS code, the decoder can correct simultaneously v errors and µ erasures in the received data if 2v + µ ≤n - k (correctable range) and will fail otherwise (uncorrectable range). We give detailed reviews of both Berlekamp-Massy and Continued-Fraction algorithms for error-and-erasure decoding. Berlekamp-Massy algorithm has long been known but sometimes appeared incorrectly in some references. Continued-Fraction algorithm has been recently applied for error-and-erasure decoding. Finally, we verify by simulation that two algorithms work exactly the same even in the uncorrectable range.

Almost everywhere balanced sequences of complexity 2n + 1

We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set $\{1,2\}^\mathbb{N}$ of directive sequences. For a given set $\mathcal{C}$ of two substitutions, we show that there exists a $\mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $\mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $\mu$-almost every $\mathcal{C}$-adic sequence is balanced, where $\mu$ is any shift-invariant ergodic Borel probability measure on $\{1,2\}^\mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $\mu$ is negative.

On Gauss--Kuzmin statistics and the transfer operator for a multidimensional continued fraction algorithm: the triangle map

On Gauss--Kuzmin statistics and the transfer operator for a multidimensional continued fraction algorithm: the triangle map

Odd-odd continued fraction algorithm

By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among others, it is proved that all the best approximating rationals of odd denominators and odd numerators of an irrational number are given by the principal convergents of the odd-odd continued fraction algorithm and vice versa.

Ergodicity of Iwasawa continued fractions via markable hyperbolic geodesics

AbstractWe prove the convergence and ergodicity of a wide class of real and higher-dimensional continued fraction algorithms, including folded and$\alpha $-type variants of complex, quaternionic, octonionic, and Heisenberg continued fractions, which we combine under the framework of Iwasawa continued fractions. The proof is based on the interplay of continued fractions and hyperbolic geometry, the ergodicity of geodesic flow in associated modular manifolds, and a variation on the notion of geodesic coding that we refer to as geodesic marking. As a corollary of our study of markable geodesics, we obtain a generalization of Serret’s tail-equivalence theorem for almost all points. The results are new even in the case of some real and complex continued fractions.

Security Issues of Novel RSA Variant

The RSA is one of the current default cryptosystems that provides security with applications such as encryptions and digital signatures. It is important to further study the weak characteristics of the RSA to ensure correct utilisation in order not to be susceptible to adversaries. In this paper, we give detailed analysis on security of the Murru-Saettone variant of the RSA cryptosystem that utilised a cubic Pell ed – k ( p2 + p + 1 ) ( q2 + q + 1 ) = 1 as key equation and N = pq as RSA modulus. We propose some attacks on this variant when the prime difference |p – q| is small. Our first approach is to utilise the continued fractions algorithm to determine the parameter d which enables us to determine the secret p and q. Our second approach considers the Coppersmith’s method and lattice basis reduction to factor the modulus N. Our attacks improve recent cryptanalyses on the cubic Pell equation variant of RSA. Furthermore, our attacks prove that under small prime difference scenario, the number of susceptible private exponents for the cubic Pell equation variant of RSA is much larger than the standard RSA.

Increment of insecure RSA private exponent bound through perfect square RSA diophantine parameters cryptanalysis

The public parameters of the RSA cryptosystem are represented by the pair of integers N and e. In this work, first we show that if e satisfies the Diophantine equation of the form ex2−ϕ(N)y2=z for appropriate values of x,y and z under certain specified conditions, then one is able to factor N. That is, the unknown yx can be found amongst the convergents of eN via continued fractions algorithm. Consequently, Coppersmith’s theorem is applied to solve for prime factors p and q in polynomial time. We also report a second weakness that enabled us to factor k instances of RSA moduli simultaneously from the given (Ni,ei) for i=1,2,⋯,k and a fixed x that fulfills the Diophantine equation eix2−yi2ϕ(Ni)=zi. This weakness was identified by solving the simultaneous Diophantine approximations using the lattice basis reduction technique. We note that this work extends the bound of insecure RSA decryption exponents.

Hyperbolic Jigsaws and Families of Pseudomodular Groups II

Abstract In our previous paper [ 5], we introduced a hyperbolic jigsaw construction and constructed infinitely many noncommensurable, nonuniform, non-arithmetic lattices of $\textrm{PSL}(2,{\mathbb R})$ with cusp set $\mathbb{Q} \cup \{\infty \}$ (called pseudomodular groups by Long and Reid [ 4]), thus answering a question posed by Long and Reid. In this paper, we continue with our study of these jigsaw groups exploring questions of arithmeticity, pseudomodularity, and also related pseudo-Euclidean and continued fraction algorithms arising from these groups. We also answer another question of Long and Reid [ 4] by demonstrating a recursive formula for the tessellation of the hyperbolic plane arising from Weierstrass groups, which generalizes the well-known “Farey addition” used to generate the Farey tessellation.

Periodic representations for quadratic irrationals in the field of 𝑝-adic numbers

Continued fractions have been widely studied in the field of p p -adic numbers Q p \mathbb Q_p , but currently there is no algorithm replicating all the good properties that continued fractions have over the real numbers regarding, in particular, finiteness and periodicity. In this paper, first we propose a periodic representation, which we will call standard, for any quadratic irrational via p p -adic continued fractions, even if it is not obtained by a specific algorithm. This periodic representation provides simultaneous rational approximations for a quadratic irrational both in R \mathbb R and Q p \mathbb Q_p . Moreover given two primes p 1 p_1 and p 2 p_2 , using the Binomial transform, we are also able to pass from approximations in Q p 1 \mathbb {Q}_{p_1} to approximations in Q p 2 \mathbb {Q}_{p_2} for a given quadratic irrational. Then, we focus on a specific p p –adic continued fraction algorithm proving that it stops in a finite number of steps when processes rational numbers, solving a problem left open in a paper by Browkin [Math. Comp. 70 (2001), pp. 1281–1292]. Finally, we study the periodicity of this algorithm showing when it produces standard representations for quadratic irrationals.

Analysis of generalized continued fraction algorithms over polynomials

We study and compare natural generalizations of Euclid's algorithm for polynomials with coefficients in a finite field. This leads to gcd algorithms together with their associated continued fraction maps. The gcd algorithms act on triples of polynomials and rely on two-dimensional versions of the Brun, Jacobi–Perron and fully subtractive continued fraction maps, respectively. We first provide a unified framework for these algorithms and their associated continued fraction maps. We then analyse various costs for the gcd algorithms, including the number of iterations and two versions of the bit-complexity, corresponding to two representations of polynomials (the usual and the sparse one). We also study the associated two-dimensional continued fraction maps and prove the invariance and the ergodicity of the Haar measure. We deduce corresponding estimates for the costs of truncated trajectories under the action of these continued fraction maps, obtained thanks to their transfer operators, and we compare the two models (gcd algorithms and their associated continued fraction maps). Proving that the generating functions appear as dominant eigenvalues of the transfer operator allows indeed a fine comparison between the models.

Ergodicity for [formula omitted]-adic continued fraction algorithms

Following Schweiger’s generalization of multidimensional continued fraction algorithms, we consider a very large family of p-adic multidimensional continued fraction algorithms, which include Schneider’s algorithm, Ruban’s algorithms, and the p-adic Jacobi–Perron algorithm as special cases. The main result is to show that all the transformations in the family are ergodic with respect to the Haar measure.