We prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by F. Schweiger and studied also by C. Kraaikamp and A. Lopes. Our main result is proven following the strategy used by Ya. Sinai and C. Ulcigrai in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.
In this paper we prove the following renewal-type limit theorem. Given an irrational $α$ in (0,1) and R>0, let $q_{n_R}$ be the first denominator of the convergents of $α$ which exceeds R. The main result in the paper is that the ratio $q_{n_R}/R$ has a limiting distribution as R tends to infinity. The existence of the limiting distribution uses mixing of a special flow over the natural extension of the Gauss map.
We prove results concerning the joint limiting distribution of the renewal time of denominators and consecutive digits of random irrational numbers in the case of continued fractions with even partial quotients, with odd partial quotients, and for Nakada's α-expansions.
We investigate from a multifractal analysis point of view the increasing rate of the sums of partial quotients $S\_n(x)=\sum\_{j=1}^n a\_j(x)$, where $x=[a\_1(x), a\_2(x), \cdots ]$ is the continued fraction expansion of an irrational $x\in (0,1)$. Precisely, for an increasing function $φ: \mathbb{N} \rightarrow \mathbb{N}$, one is interested in the Hausdorff dimension of the sets\[E\_φ= \left\{x\in (0,1): \lim\_{n\to\infty} \frac {S\_n(x)} {φ(n)} =1\right\}.\]Several cases are solved by Iommi and Jordan, Wu and Xu, and Xu. We attack the remaining subexponential case $\exp(n^γ), \ γ\in [1/2, 1…
We consider sets of real numbers in $[0,1)$ with prescribed frequencies of partial quotients in their regular continued fraction expansions. It is shown that the Hausdorff dimensions of these sets, always bounded from below by $1/2$, are given by a modified variational principle.
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 …
The classical Khintchine and Jarník theorems, generalizations of a consequence of Dirichlet's theorem, are fundamental results in the theory of Diophantine approximation. These theorems are concerned with the size of the set of real numbers for which the partial quotients in their continued fraction expansions grows with a certain rate. Recently it was observed that the growth of product of pairs of consecutive partial quotients in the continued fraction expansion of a real number is associated with improvements to Dirichlet's theorem. In this paper we consider the products of several consecut…
In this paper, we provide a model for cross sections to the geodesic and horocycle flows on $\operatorname{SL}(2, \mathbb{R})/G_q$ using an extension of a heuristic of P. Arnoux and A. Nogueira. Our starting point is a continued fraction algorithm related to the group $G_q$, and a cross section to the horocycle flow on $\operatorname{SL}(2, \mathbb{R})/G_q$ from a previous paper. As an application, we get the natural extension and invariant measure for a symmetric $G_q$-Farey interval map resulting from projectivizing the aforementioned continued fraction algorithm.
We consider an analogue of Nakada's $α$-continued fraction transformation in the setting of continued fractions with odd partial quotients. More precisely, given $α\in [\frac{1}{2}(\sqrt{5}-1),\frac{1}{2}(\sqrt{5}+1)]$, we show that every irrational number $x\in I_α=[α-2,α)$ can be uniquely represented as $$ x= \cfrac{e_1 (x;α)}{d_1 (x;α) +\cfrac{e_2(x;α)}{d_2(x;α)+\cdots}} , $$ with $e_i(x;α) \in \{ \pm 1\}$ and $d_i(x;α) \in 2{\mathbb N} -1$ determined by the iterates of the transformation $$φ_α(x) := \frac{1}{| x|} - 2 \bigg[ \frac{1}{2| x|} +\frac{1-α}{2} \bigg]-1$$ of $I_α$. We also descr…
We present a detailed Hausdorff dimension analysis of the set of real numbers where the product of consecutive partial quotients in their continued fraction expansion grow at a certain rate but the growth of the single partial quotient is at a different rate. We consider the set \begin{equation*} \FF(Φ_1,Φ_2) \defeq \EE(Φ_1) \backslash \EE(Φ_2)=\left\{x\in[0,1): \begin{split} a_n(x)a_{n+1}(x) & \geqΦ_1(n) \text{\,\, for infinitely many } n\in\N a_{n+1}(x) & <Φ_2(n) \text{\,\, for all sufficiently large } n\in\N \end{split} \right\}, \end{equation*} where $Φ_i:\N\to(0,\infty)$ are any functions…
For $x\in [0,1)$, let $x=[a_1(x), a_2(x),...]$ be its continued fraction expansion with partial quotients ${a_n(x), n\ge 1}$. Let $ψ: \mathbb{N} \rightarrow \mathbb{N}$ be a function with $ψ(n)/n\to \infty$ as $n\to \infty$. In this note, the fast Khintchine spectrum, i.e., the Hausdorff dimension of the set $$ E(ψ):=\Big{x\in [0,1): \lim_{n\to\infty}\frac{1}{ψ(n)}\sum_{j=1}^n\log a_j(x)=1\Big} $$ is completely determined without any extra condition on $ψ$.
Motivated by recent developments in the metrical theory of continued fractions for real numbers concerning the growth of consecutive partial quotients, we consider its analogue over the field of formal Laurent series. Let $A_n(x)$ be the $n$th partial quotient of the continued fraction expansion of $x$ in the field of formal Laurent series. We consider the sets of $x$ such that $°A_{n+1}(x)+\cdots+°A_{n+k}(x)~\ge~Φ(n)$ holds for infinitely many $n$ and for all $n$ respectively, where $k\ge1$ is an integer and $Φ(n)$ is a positive function defined on $\mathbb{N}$. We determine the size of these…
Let $[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0, 1)$. The study of the growth rate of the product of consecutive partial quotients $a_n(x)a_{n+1}(x)$ is associated with the improvements to Dirichlet's theorem (1842). We establish both the weak and strong laws of large numbers for the partial sums $S_n(x)= \sum_{i=1}^n a_i(x)a_{i+1}(x)$ as well as, from a multifractal analysis point of view, investigate its increasing rate. Specifically, we prove the following results: \medskip \begin{itemize} \item For any $ε>0$, the Lebesgue me…
Two closely related families of $α$-continued fractions were introduced in 1981: by Nakada on the one hand, by Tanaka and Ito on the other hand. The behavior of the entropy as a function of the parameter $α$ has been studied extensively for Nakada's family, and several of the results have been obtained exploiting an algebraic feature called matching. In this article we show that matching occurs also for Tanaka-Ito $α$-continued fractions, and that the parameter space is almost completely covered by matching intervals. Indeed, the set of parameters for which the matching condition does not hold…
The Lévy constant of an irrational real number is defined by the exponential growth rate of the sequence of denominators of the principal convergents in its continued fraction expansion. Any quadratic irrational has an ultimately periodic continued fraction expansion and it is well-known that this implies the existence of a Lévy constant. Let $a, b$ be distinct positive integers. If the sequence of partial quotients of an irrational real number is a Sturmian sequence over $\{a, b\}$, then it has a Lévy constant, which depends only on $a$, $b$, and the slope of the Sturmian sequence, but not on…
We construct an absolutely normal number whose continued fraction expansion is normal in the sense that it contains all finite patterns of partial quotients with the expected asymptotic frequency as given by the Gauss-Kuzmin measure. The construction is based on ideas of Sierpinski and uses a large deviations theorem for sums of mixing random variables.
Extreme value theorem for geodesic flow on the quotient of the theta group
HUN-REN Alfréd Rényi Institute of Mathematics（HUN-REN 阿尔弗雷德·雷尼数学研究所）；Department of Mathematical Sciences, Seoul National University（首尔大学数学科学系）；Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）
For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients is a prefix of that of $x$. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set $E(ψ)$ of complex numbers which are well approximated with the given bound $ψ$ and have quite different Hurwitz contin…
We study analogues of Minkowski's question mark function $?(x)$ related to continued fractions with even or odd partial quotients. We prove that these functions are Hölder continuous with precise exponents, and that they linearize the appropriate versions of the Gauss and Farey maps.
How many points contain homothetic copies in their Hurwitz continued fraction expansion?
Faculty of Science and Engineering, Doshisha University（立命馆大学理工学部）；Keio Institute of Pure and Applied Sciences (KiPAS), Department of Mathematics, Keio University（庆应义塾大学纯理应用科学研究所（KiPAS）数学系）
Expansions that furnish increasingly good approximations to real numbers are usually related to dynamical systems. Although comparing dynamical systems seems difficult in general, Lochs was able in 1964 to relate the relative speed of approximation of decimal and regular continued fraction expansions (almost everywhere) to the quotient of the entropies of their dynamical systems. He used detailed knowledge of the continued fraction operator. In 2001, a generalization of Lochs' result was given by Dajani and Fieldsteel in \citeDajF, describing the rate at which the digits of one number-theoreti…
We consider the ensemble of curves $\{γ_{α,N}:α\in(0,1],N\in\N\}$ obtained by linearly interpolating the values of the normalized theta sum $N^{-1/2}\sum_{n=0}^{N'-1}\exp(πi n^2α)$, $0\leq N'<N$. We prove the existence of limiting finite-dimensional distributions for such curves as $N\to\infty$, with respect to an absolutely continuous probability measure $μ_R$ on $(0,1]$. Our Main Theorem generalizes a result by Marklof and Jurkat and van Horne. Our proof relies on the analysis of the geometric structure of such curves, which exhibit spiral-like patterns (curlicues) at different scales. We ex…
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 …
Using techniques introduced by D. Mayer, we prove an extension of the classical Gauss-Kuzmin theorem about the distribution of continued fractions, which in particular allows one to take into account some congruence properties of successive convergents. This result has an application to the Mixmaster Universe model in general relativity. We then study some averages involving modular symbols and show that Dirichlet series related to modular forms of weight 2 can be obtained by integrating certain functions on real axis defined in terms of continued fractions. We argue that the quotient $PGL(2,\…
We initiate the study of the sets $H(c)$, $0<c<1$, of real $x$ for which the sequence $(kx)_{k\geq1}$ (viewed mod 1) consistently hits the interval $[0,c)$ at least as often as expected (i. e., with frequency $\geq c$). More formally, \[ H(c)=\{α\in \mathbf R\mid {\rm card}(\{1\leq k\leq n\mid < kα><c\})\geq cn, {for all}n\geq1\}. \] where $<x>=x-[x]$ stands for the fractional part of $x\in \mathbb R$. We prove that, for rational $c$, the sets $H(c)$ are of positive Hausdorff dimension and, in particular, are uncountable. For integers $m\geq1$, we obtain a surprising characterization of the nu…
Let $\{a_n\}_1^\infty$ and $\{θ_n\}_0^\infty$ be the sequences of partial quotients and approximation coefficients for the continued fraction expansion of an irrational number. We will provide a function $f$ such that $a_{n+1} = f(θ_{n\pm1},θ_n)$. In tandem with a formula due to Dajani and Kraaikamp, we will write $θ_{n \pm 1}$ as a function of $(θ_{n \mp 1}, θ_n)$, revealing an elegant symmetry in this classical sequence and allowing for its recovery from a pair of consecutive terms.
