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There are two main interrelated goals of this paper. Firstly we investigate the sums

We 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.

We develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, we develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen’s parameter. We illustrate our results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.

We determine all pairs ðp;nÞ,wherep is a prime and n a positive integer, such that there exists a reduced fraction u=v > 1 with uþv ¼ n and u=v has a nonterminating Schneider's p-adic continued fraction expansion. We also prove a bound on the length of the preperiodinthep-adic continued fraction of u=v when maxfjuj;jvjg

The paper is concerned with continued fractions having bounded even-order partial quotients. We demonstrate that every real number can be written as a sum of two continued fractions whose even-order partial quotients are equal to 1, and every positive number can be written as a product of two such continued fractions. Then we study the Hausdorff dimension of the set of continued fractions whose even-order partial quotients are all equal to a given positive integer c. Taking in particular c=1, we show that the set of continued fractions with even-order partial quotients equal to 1 has the Hausdorff dimension between 0.732 and 0.819.

Multidimensional continued fractions (MCFs) were introduced by Jacobi and Perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the p-adic numbers \\mathbb{Q}_p. First, we introduce them from a formal point of view, i.e., without considering a specific algorithm that produces the partial quotients of an MCF, and we perform a general study about their convergence in \\mathbb{Q}_p. In particular, we derive some sufficient conditions for their convergence and we prove that convergent MCFs always strongly converge in \\mathbb{Q}_p contrary to the real case where strong convergence is not always guaranteed. Then, we focus on a specific algorithm that, starting from an m-tuple of numbers in \\mathbb{Q}_p (p odd), produces the partial quotients of the corresponding MCF. We see that this algorithm is derived from a generalized p-adic Euclidean algorithm and we prove that it always terminates in a finite number of steps when it processes rational numbers.

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.

We study branched continued fractions with inequivalent variables, branched continued fractions of the special form, and multidimensional C - and S -fractions with inequivalent variables. By using the results established for continued fractions and the results concerning the convergence and estimation of the errors of approximation of branched continued fractions of a special form in angular domains, we obtain new estimates for the rate of convergence of branched continued fractions of a special form, pointwise convergence of multidimensional C -fractions, and uniform convergence on compact sets of angular domains of multidimensional S -fractions with inequivalent variables.

The paper discusses the prospect of using a combined model based on finite segments (polynomials) of the Volterra integral power series. We consider a case when the problem of identifying the Volterra kernels is solved. The predictive properties of the classic Volterra polynomial are improved by adding a linear part in the form of an equivalent continued fraction. This technique allows us to distinguish an additional parameter—the connection coefficient α, which is effective in adapting the constructed integral model to changes in technical parameters at the input of a dynamic system. In addition, this technique allows us to take into account the case of perturbing the kernel of the linear term of the Volterra polynomial in the metric C[0,T] by a given value δ, implying the ideas of Volterra regularizing procedures. The problem of choosing the connection coefficient is solved using a special extremal problem. The developed algorithms are used to solve the problem of identifying input signals of test dynamic systems, among which, in addition to mathematical ones, thermal power engineering devices are used.

We give a new class of multidimensional p-adic continued fraction algorithms. We propose an algorithm in the class for which we can expect that the multidimensional p-adic version of Lagrange’s Theorem will hold.