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Recently a new class of continued fraction algorithms, the (N , α )-expansions, was introduced in Kraaikamp and Langeveld (J Math Anal Appl 454(1):106–126, 2017) for each N ∈ N , N ≥ 2 and α ∈ (0 , N - 1 ] . Each of these continued fraction algorithms has only finitely many possible digits. These (N , α) -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 small enough, all rationals have an eventually periodic expansion with period 1. This happens for all α when N = 2 . We also find infinitely many matching intervals for N = 2 , as well as rationals that are not contained in any matching interval. [ABSTRACT FROM AUTHOR]

This paper deals with quadratic irrationals of the form m / q + v for fixed positive integers v and q, v not a square, and varying integers m, (m , q) = 1 . Two numbers m / q + v , n / q + v of this kind are equivalent (in a classical sense) if their continued fraction expansions can be written with the same period. We give a necessary and sufficient condition for the equivalence in terms of solutions of Pell's equation. Moreover, we determine the number of equivalence classes to which these quadratic irrationals belong. [ABSTRACT FROM AUTHOR]

The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to p-adic numbers where it presents many differences with respect to the real case. In this paper we investigate periodicity for the p-adic continued fractions introduced by Browkin. We give some necessary and sufficient conditions for periodicity in general, although a full characterization of p-adic numbers having purely periodic Browkin continued fraction expansion is still missing. In the second part of the paper, we describe a general procedure to construct square roots of integers having periodic Browkin p-adic continued fraction expansion of prescribed even period length. As a consequence, we prove that, for every n ≥ 1 , there exist infinitely many m ∈ Q p with periodic Browkin expansion of period 2 n , extending a previous result of Bedocchi obtained for n = 1 . [ABSTRACT FROM AUTHOR]

q -deformed real numbers are power series with integer coefficients. We study Stieltjes and Jacobi type continued fraction expansions of q -deformed real numbers and find many new examples of such continued fractions. We also investigate the corresponding sequences of Hankel determinants and find an infinite family of power series for which several of the first sequences of Hankel determinants consist of − 1 , 0 and 1 only. These Hankel sequences satisfy Somos and Gale-Robinson recurrences. [ABSTRACT FROM AUTHOR]

Continued fractions have been widely studied in the field of p-adic numbers \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-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 and Qp. Moreover given two primes p1 and p2, using the Binomial transform, we are also able to pass from approximations in Qp1 to approximations in Qp2 for a given quadratic irrational. Then, we focus on a specific 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. [ABSTRACT FROM AUTHOR]

In this paper, we study Hurwitz complex continued fraction (HCCF) expansion of some complex quadratic irrationals. We observed that some complex quadratic irrationals can be expanded into two HCCFs. On comparing the Hurwitz complex continued fraction with simple continued fraction of any real quadratic irrational, it was found that there are differences between the two. [ABSTRACT FROM AUTHOR]

A well-known theorem of Lagrange states that the simple continued fraction of a real number is periodic if and only if is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two interlacing geometric series and show that in certain cases they converge to quadratic irrationalities. This phenomenon is connected with certain sequences of polynomials whose properties we examine further. [ABSTRACT FROM AUTHOR]

We study the quadratic Lagrange spectrum defined by Parkkonen and Paulin by considering the approximation by quadratic numbers whose regular continued fraction expansion is ultimately periodic with the same period as a fixed quadratic number or its Galois conjugate. We improve the upper bound on the approximation constants involved thereby proving a conjecture stated by Bugeaud. [ABSTRACT FROM AUTHOR]

Abstract: We present an analysis of autocorrelation functions in symmetric barrier billiards using a renormalisation approach for quadratic irrational trajectories. Depending on the nature of the barrier, this leads to either self-similar or chaotic behaviour. In the self-similar case we give an analysis of the half barrier and present a detailed calculation of the locations, asymptotic heights and signs of the main peaks in the autocorrelation function. Then we consider arbitrary barriers, illustrating that typically these give rise to chaotic correlations of the autocorrelation function which we further represent by showing the invariant sets associated with these correlations. Our main ingredient here is a functional recurrence which has been previously derived and used in work on the Harper equation, strange non-chaotic attractors and a quasi-periodically forced two-level system. [Copyright &y& Elsevier]

Abstract: The continued fraction expansion of a real number may be studied by considering a suitable discrete dynamical system of dimension two. In the special case where the number to be expanded is a quadratic irrational, that is a positive irrational root of a polynomial of degree two, more insight may be gained by considering a new dynamical system of dimension three, where the state vector stores the coefficients of the quadratic polynomials resulting from the expansion process. We show that a number of constants of motions can be derived and exploited to explore the attracting set of the solutions. Links with the solution to Pell’s equations are also investigated. [Copyright &y& Elsevier]

We develop further the theory of q -deformations of real numbers introduced in [10] and [9] and focus in particular on the class of real quadratic irrationals. Our key tool is a q -deformation of matrices of the modular group PSL (2 , Z). The action of the modular group by Möbius transformations commutes with the q -deformations. We prove that the traces of the q -deformed matrices are palindromic polynomials with positive coefficients. These traces appear in the explicit expressions of the q -deformed quadratic irrationals. [ABSTRACT FROM AUTHOR]

Some characteristic approximation properties of quadratic irrationals are studied in this paper. It is shown that the limit points of the sequence δn form a subset C(x), and D(x) can be generated from C(x) in a relatively simple way. Another proof of Lekkerkerker's theorem is given using relations between δn−1, δn, δn+1 which are independent of x and n.

Abstract: Quadratic irrationals and their representation as continued fractions are investigated by means of proper tools of Discrete Dynamical Systems theory. This is done by recasting the process of generation of digits in the continued fraction expansion of a real number as a suitable discrete nonlinear system whose asymptotic behavior is to be studied. Such an approach allows, on the one hand, to retrieve well known results in the literature, the most relevant being the Lagrange theorem, and on the other hand it aims to give more insight into some related issues. [Copyright &y& Elsevier]