arXiv subjects

Cor Kraaikamp

Publications and source records attributed to Cor Kraaikamp.

At least 19 recordsLinked to original sources

arXiv2026

Sharpening Borel's result in Diophantine approximation

In this paper, we refine Borel's 1903 result in Diophantine approximation by providing sharper bounds for the minimum of three consecutive approximation coefficients $\Theta_n(x)$, defined for any real number $x$ with regular continued fraction (RCF) expansion $x=[0;a_1,a_2,\dots]$ as $\Theta_n = q_n^2\left| x-\frac{p_n}{q_n}\right|$. Here $\frac{p_n}{q_n}$ is the $n$th RCF convergent of $x$. Borel's result states that for all (irrational) $x$ and all $n\in\mathbb{N}$, $$ \min \left\{ \Theta_{n-1}(x),\Theta_n(x),\Theta_{n+1}(x)\right\} \leq \frac{1}{\sqrt{5}}. $$ We focus on the situation where $a_{n+1}=1$, since otherwise a result by F.~Bagemihl and J.R.~McLaughlin from 1966 implies that the Borel-bound $1/\sqrt{5}$ can already be improver to $1/\sqrt{8}$.

math.DS↗

arXiv2026

A strange continued fraction associated with the Romik map

In 2008, Dan Romik studied in this journal Primitive Pythagorean Triples, or PPTs. In order to do so, he introduced a modified slow (subtractive) Euclidean algorithm, and showed that the underlying dynamical system of this Euclidean algorithm (the ``Romik system''), is ergodic and has a $\sigma$-finite, infinite measure, of which is explicitly given. In this paper, the Romik system is further studied. Various basic properties are determined, such as the expansion of rational numbers and quadratic irrationals. Also (a version of) the planar natural extension of the Romik system is obtained, and the $\sigma$-finite, invariant measure is explicitly given, and it is shown that it is ergodic. Furthermore, for Lebesgue almost every $x$ asymptotically half of the regular continued fraction (RCF) convergents of $x$ are among the Romik convergents. We also show that related to the Romik map a ``strange'' continued fraction can be given. ``Strange,'' as the set of possible partial quotients (i.e., digits) for any $x\in [0,1]$ in this expansion is $\{ 0, \pm 2\}$. Various properties of this ``Romik expansion'' are given.

math.DS↗

arXiv2025

Sharpening Vahlen's result in Diophantine approximation

n this paper we refine Vahlen's 1895 result in Diophantine approximation by providing sharper bounds for the approximation coefficients, especially when at least one of the partial quotients $a_n$ or $a_{n+1}$ of the regular continued fraction expansion $[a_0;a_1,a_2,\dots]$ of $x$ is 1. An improvement of Vahlen's result was already given in papers by Jaroslav Hanucl ([9]), Hanucl and Silvie Bahnerova ([10]), and by Dinesh Sharma Bhattarai ([5]), but the approach of the present paper is very different from Hanucl c.s. We believe that the geometrical methods used in this paper not only offer a significant improvement over Vahlen's result, but also yield new insights that can contribute to improving Borel's classical constant.

math.DS↗

arXiv2025

Inducing contractions of the mother of all continued fractions

We introduce a new, large class of continued fraction algorithms producing what are called contracted Farey expansions. These algorithms are defined by coupling two acceleration techniques -- induced transformations and contraction -- in the setting of Shunji Ito's natural extension of the Farey tent map, which generates `slow' continued fraction expansions. In addition to defining new algorithms, we also realise several existing continued fraction algorithms in our unifying setting. In particular, we find regular continued fractions, the second-named author's $S$-expansions, and Nakada's parameterised family of $α$-continued fractions for all $0<α\le 1$ as examples of contracted Farey expansions. Moreover, we give a new description of a planar natural extension for each of the $α$-continued fraction transformations as an explicit induced transformation of Ito's natural extension.

math.NT↗

arXiv2024

Proofs of ergodicity of piecewise Möbius interval maps using planar extensions

We give two results for deducing dynamical properties of piecewise Möbius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both follow from mild finiteness conditions on the planar extension along with a new property ``bounded non-full range" used to relax traditional Markov conditions. Second, the ``quilting" operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise Möbius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada $α$-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups.

math.DS↗

arXiv2023

A unifying theory for metrical results on regular continued fraction convergents and mediants

We revisit Ito's (\cite{I1989}) natural extension of the Farey tent map, which generates all regular continued fraction convergents and mediants of a given irrational. With a slight shift in perspective on the order in which these convergents and mediants arise, this natural extension is shown to provide an elegant and powerful tool in the metric theory of continued fractions. A wealth of old and new results -- including limiting distributions of approximation coefficients, analogues of a theorem of Legendre and their refinements, and a generalisation of L\'evy's Theorem to subsequences of convergents and mediants -- are presented as corollaries within this unifying theory.

math.DS↗

arXiv2023

(non)-matching and (non)-periodicity for $(N,α)$-expansions

Recently a new class of continued fraction algorithms, the $(N,α$)-expansions, was introduced for each $N\in\mathbb{N}$, $N\geq 2$ and $α\in (0,\sqrt{N}-1]$. 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. 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.

math.DS↗

arXiv2023

Continuity of entropy for all $α$-deformations of an infinite class of continued fraction transformations

We extend the results of our 2020 paper in the Annali della Scuola Normale Superiore di Pisa, Classe di Scienze. There, we associated to each of an infinite family of triangle Fuchsian groups a one-parameter family of continued fraction maps and showed that the matching (or, synchronization) intervals are of full measure. Here, we find planar extensions of each of the maps, and prove the continuity of the entropy function associated to each one-parameter family. We also introduce a notion of "first pointwise expansive power" of an eventually expansive interval map. We prove that for every map in one of our one-parameter families its first pointwise expansive power map has its natural extension given by the first return of the geodesic flow to a cross section in the unit tangent bundle of the hyperbolic orbifold uniformized by the corresponding group. We conjecture that this holds for all of our maps. We give numerical evidence for the conjecture.

math.DS↗

arXiv2022

Matching of orbits of certain $N$-expansions with a finite set of digits

In this paper we consider a class of continued fraction expansions: the so-called $N$-expansions with a finite digit set, where $N\geq 2$ is an integer. These \emph{$N$-expansions with a finite digit set} were introduced in [KL,L], and further studied in [dJKN,S]. For $N$ fixed they are steered by a parameter $α\in (0,\sqrt{N}-1]$. In [KL], for $N=2$ an explicit interval $[A,B]$ was determined, such that for all $α\in [A,B]$ the entropy $h(T_α)$ of the underlying Gauss-map $T_α$ is equal. In this paper we show that for all $N\in \mathbb N$, $N\geq 2$, such plateaux exist. In order to show that the entropy is constant on such plateaux, we obtain the underlying planar natural extension of the maps $T_α$, the $T_α$-invariant measure, ergodicity, and we show that for any two $α,α'$ from the same plateau, the natural extensions are metrically isomorphic, and the isomorphism is given explicitly. The plateaux are found by a property called matching.

math.DS↗

arXiv2021

Natural extensions and entropy of $α$-continued fraction expansions with odd partial quotients

In an article (which we will refer to as [BM]) of Boca and the fourth author of this paper, a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter $α\in [g,G]$, where $g=\tfrac{1}{2}(\sqrt{5}-1)$ and $G=g+1=1/g$ are the two golden mean numbers is introduced. In this article, by using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [BM] are obtained, and it is shown that for each $α,α^*\in [g,G]$ the natural extensions from [BM] are metrically isomorphic. An immediate consequence of this is, that the entropy of all these natural extensions is equal for $α\in [g,G]$, a fact already observed in [BM]. Furthermore, it is shown that this approach can be extended to values of $α$ smaller than $g$, and that for values of $α\in [\tfrac{1}{6}(\sqrt{13}-1), g]$ all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of $α\in [0,G]$. It is shown that in any neighborhood of $0$ we can find intervals on which the entropy is decreasing, intervals on which the entropy is increasing and intervals on which the entropy is constant. In order to prove this we use a phenomena called matching.

math.DS↗

arXiv2017

Natural Extensions for Nakada's alpha-expansions: descending from 1 to g^2

By means of singularisations and insertions in Nakada's alpha-expansions, which involves the removal of partial quotients 1 while introducing partial quotients with a minus sign, the natural extension of Nakada's continued fraction map T_alpha is given for (\sqrt{10}-2)/3\leqα<1. From our construction it follows that Ω_α, the domain of the natural extension of T_α, is metrically isomorphic to Ω_g for α\in [g^2,g), where g is the small golden mean. Finally, although Ω_αproves to be very intricate and unmanageable for α\in [g^2, (\sqrt{10}-2)/3), the α-Legendre constant L(α) on this interval is explicitly given.

math.DS↗

arXiv2017

Synchronization is full measure for all $α$-deformations of an infinite class of continued fraction transformations

We study an infinite family of one-parameter deformations, so-called $α$-continued fractions, of interval maps associated to distinct triangle Fuchsian groups. In general for such one-parameter deformations, the function giving the entropy of the map indexed by $α$ varies in a way directly related to whether or not the orbits of the endpoints of the map synchronize. For two cases of one-parameter deformations associated to the classical case of the modular group $\text{PSL}_2(\mathbb Z)$, the set of $α$ for which synchronization occurs has been determined. Here, we explicitly determine the synchronization sets for each $α$-deformation in our infinite family. (In general, our Fuchsian groups are not subgroups of the modular group, and hence the tool of relating $α$-expansions back to regular continued fraction expansions is not available to us.) A curiosity here is that all of our synchronization sets can be described in terms of a single tree of words. In a paper in preparation, we identify the natural extensions of our maps, as well as the entropy functions associated to each deformation.

math.DS↗

arXiv2016

Hankel Matrices for the Period-Doubling Sequence

We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding Hankel matrices. Similar considerations give the Hankel determinants for other orders.

math.CO↗

arXiv2016

Invariant measures for continued fraction algorithms with finitely many digits

In this paper we consider continued fraction (CF) expansions on intervals different from $[0,1]$. For every $x$ in such interval we find a CF expansion with a finite number of possible digits. Using the natural extension, the density of the invariant measure is obtained in a number of examples. In case this method does not work, a Gauss-Kuzmin-Lévy based approximation method is used. Finally, a subfamily of the $N$-expansions is studied. In particular, the entropy as a function of a parameter $α$ is estimated for $N=2$ and $N=36$. Interesting behavior can be observed from numerical results.

math.DS↗

arXiv2012

Natural extensions and entropy of $α$-continued fractions

We construct a natural extension for each of Nakada's $α$-continued fractions and show the continuity as a function of $α$ of both the entropy and the measure of the natural extension domain with respect to the density function $(1+xy)^{-2}$. In particular, we show that, for all $0 < α\le 1$, the product of the entropy with the measure of the domain equals $π^2/6$. As a key step, we give the explicit relationship between the $α$-expansion of $α-1$ and of $α$.

math.DS↗

arXiv2011

Understanding the non-Gaussian nature of reactive solute transport. From particle dynamics to the partial differential equations

In the present study we examine non-Gaussian spreading of solutes subject to advection, dispersion and kinetic sorption (adsorption/desorption). We start considering the behavior of a single particle and apply a random walk to describe advection/dispersion plus a Markov chain to describe kinetic sorption. We show in a rigorous way that this model leads to a set of differential equations. For this combination of stochastic processes such a derivation is new. Then, to illustrate the mechanism that leads to non-Gaussian spreading we analyze this set of equations at first leaving out the Gaussian dispersion term (microdispersion). The set of equations now transforms to the telegrapher's equation. Characteristic for this system is a longitudinal spreading, that becomes Gaussian only in the long-time limit. We refer to this as kinetics induced spreading. When the microdispersion process is included back again, the characteristics of the telegraph equations are still present. Now two spreading phenomena are active, the Gaussian microdispersive spreading plus the kinetics induced non-Gaussian spreading. In the long run the latter becomes Gaussian as well. Another non-Gaussian feature shows itself in the 2D situation. Here, the lateral spread and the longitudinal displacement are no longer independent, as should be the case for a 2D Gaussian spreading process. In a displacing plume this interdependence is displayed as a `tailing' effect. We also analyze marginal and conditional moments, which confirm this result. With respect to effective properties (velocity and dispersion) we conclude that effective parameters can be defined properly only for large times (asymptotic times). In the two-dimensional case it appears that the transverse spreading depends on the longitudinal coordinate. This results in `cigar-shaped' contours.

math.PR↗

arXiv2009

Approximation Results for alpha-Rosen Fractions

In this article we generalize Borel's classical approximation results for the regular continued fraction expansion to the alpha-Rosen fraction expansion, using a geometric method. We give a Haas-Series-type result about all possible good approximations for the alpha for which the Legendre constant is larger than the Hurwitz constant.

math.NT↗

arXiv2009

Sharp bounds for symmetric and asymmetric Diophantine approximation

In 2004, J.C. Tong found bounds for the approximation quality of a regular continued fraction convergent of a rational number, expressed in bounds for both the previous and next approximation. We sharpen his results with a geometric method and give both sharp upper and lower bounds. We also calculate the asymptotic frequency that these bounds occur.

math.NT↗