The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we get several results concerning extremes of nearest integer continued fractions as well.
We study the density of the invariant measure of the Hurwitz complex continued fraction from a computational perspective. It is known that this density is piece-wise real-analytic and so we provide a method for calculating the Taylor coefficients around certain points and also the results of our calculations. While our method does not find a simple "closed form" for the density of the invariant measure (if one even exists), our work leads us to some new conjectures about the behavior of the density at certain points. In addition to this, we detail all admissible strings of digits in the Hurwit…
Hurwitz (1887) defined a continued fraction algorithm for complex numbers which is better behaved in many respects than a more "natural" extension of the classical continued fraction algorithm to the complex plane would be. Although the Hurwitz complex continued fraction algorithm is not "reducible" to another complex continued fraction algorithm, over the reals the story is different. In this note we make clear the relation between the restriction of Hurwitz's algorithm to the real numbers and the classical continued fraction algorithm. As an application we reprove the main result of Choudhur…
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…
Good's Theorem for regular continued fraction states that the set of real numbers $[a_0;a_1,a_2,\ldots]$ such that $\displaystyle\lim_{n\to\infty} a_n=\infty$ has Hausdorff dimension $\tfrac{1}{2}$. We show an analogous result for the complex plane and Hurwitz Continued Fractions. The set of complex numbers whose Hurwitz Continued fraction $[a_0;a_1,a_2,\ldots]$ satisfies $\displaystyle\lim_{n\to\infty} |a_n|=\infty$ has Hausdorff dimension $1$, half of the ambient space's dimension.
We derive closed-form expressions for several new classes of Hurwitzian- and Tasoevian continued fractions, including $[0;\overline{p-1,1,u(a+2nb)-1,p-1,1,v(a+(2n+1)b)-1 }\,\,]_{n=0}^\infty$, $[0; \overline{c + d m^{n}}]_{n=1}^{\infty}$ and $[0; \overline{e u^{n}, f v^{ n}}]_{n=1}^\infty$. One of the constructions used to produce some of these continued fractions can be iterated to produce both Hurwitzian- and Tasoevian continued fractions of arbitrary long quasi-period, with arbitrarily many free parameters and whose limits can be determined as ratios of certain infinite series. We also deriv…
Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function
Institute of Mathematics, Vilnius University（维尔纽斯大学数学研究所）；St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院列别捷夫数学研究所圣彼得堡分部）
We prove an explicit formula for infinitely many convergents of Hurwitzian continued fractions that repeat several copies of the same constant and elements of one arithmetic progression, in a quasi-periodic fashion. The proof involves combinatorics and formal Laurent series. Using very little analysis we can express their limits in terms of (modified) Bessel functions and Fibonacci polynomials. The limit formula is a generalization of Lehmer's theorem that implies the continuous fraction expansions of $e$ and $\tan(1)$, and it can also be derived from Lehmer's work using Fibonacci polynomial i…
Adolf Hurwitz proposed in 1887 a continued fraction algorithm for complex numbers: Hurwitz continued fractions (HCF). Among other similarities between HCF and regular continued fractions, quadratic irrational numbers over $\mathbb{Q}(i)$ are precisely those with periodic HCF expansions. In this paper, we give some necessary as well as some sufficient conditions for pure periodicity of HCF. Then, we characterize badly approximable complex numbers in terms of HCF. Finally, we prove a slightly weaker complex analogue of a theorem by Y. Bugeaud on the transcendence of certain continued fractions.
A converse method to the Construction of Salem (1945) of convergent families of Salem numbers is investigated in terms of an association between Salem polynomials and Hurwitz quotients via expansive polynomials of small Mahler measure. This association makes use of Bertin-Boyd's Theorem A (1995) of interlacing of conjugates on the unit circle; in this context, a Salem number $β$ is produced and coded by an m-tuple of positive rational numbers characterizing the (SITZ) Stieltjes continued fraction of the corresponding Hurwitz quotient (alternant). The subset of Stieltjes continued fractions ove…
The Newton approximation, the Hurwitz continued fraction, and the Sierpinski series for relatively quadratic units over certain imaginary quadratic number fields
We discuss the nearest lambda_q--multiple continued fractions and their duals for lambda_q = 2 cos(pi/q) which are closely related to the Hecke triangle groups G_q, q=3,4,... . They have been introduced in the case q=3 by Hurwitz and for even q by Nakada. These continued fractions are generated by interval maps f_q respectively f_q^* which are conjugate to subshifts over infinite alphabets. We generalize to arbitrary q a result of Hurwitz concerning the G_q-- and f_q-equivalence of points on the real line. The natural extension of the maps f_q and f_q^* can be used as a Poincare map for the ge…
We prove a suite of dynamical results, including exactness of the transformation and piecewise-analyticity of the invariant measure, for a family of continued fraction systems, including specific examples over reals, complex numbers, quaternions, octonions, and in $R^3$. Our methods expand on the work of Nakada and Hensley, and in particular fill some gaps in Hensley's analysis of Hurwitz complex continued fractions. We further introduce a new ``serendipity'' condition for a continued fraction algorithm, which controls the long-term behavior of the boundary of the fundamental domain under iter…
In this work, we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin-Iosifescu asymptotics for the exceedances of digits obtained from the regular continued fraction expansion of a number chosen randomly from $(0,1)$ according to the Gauss measure. As a consequence, we significantly improve the best known upper bound on the rate of convergence of the maxima in this case. We observe that the asymptotics of order statistics and the extrema…
For $α_0 = \left[a_0, a_1, \ldots\right]$ an infinite continued fraction and $σ$ a linear fractional transformation, we study the continued fraction expansion of $σ(α_0)$ and its convergents. We provide the continued fraction expansion of $σ(α_0)$ for four general families of continued fractions and when $\left|\det σ\right| = 2$. We also find nonlinear recurrence relations among the convergents of $σ(α_0)$ which allow us to highlight relations between convergents of $α_0$ and $σ(α_0)$. Finally, we apply our results to some special and well-studied continued fractions, like Hurwitzian and Taso…
We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $|Q(x, y)| < ε$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the slopes of the lines on which $Q$ vanishes. The method is based on a coding of geodesics on the modular surface via Hurwitz expansions of the endpoints of their lifts in the Poincare half-plane.
A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry.
We define a continued fraction map associated with the $\mathfrak o(\sqrt{-3})$-module $\mathcal J = η\cdot\mathfrak o(\sqrt{-3})$, $η= \frac{3 + \sqrt{-3}}{2}$, which is an Eisenstein field version of the continued fraction map associated with $\mathfrak o(\sqrt{-1}) \cdot (1 + i)$ defined by J.~Hurwitz in the case of the Gaussian field. Together with $T$, we show that all complex numbers $z$ can be expanded as $\mathcal J$-coefficients. We discuss some basic properties of these continued fraction expansions such as the monotonicity of the absolutely value of the principal convergent $q_{n}$ …
In recent work, the first two authors constructed a generalized continued fraction called the $p$-continued fraction, characterized by the property that its convergents (a subsequence of the regular convergents) are best approximations with respect to the $L^p$ norm, where $p\geq 1$. We extend this construction to the region $0<p<1$, where now the $L^p$ quasinorm is non-convex. We prove that the approximation coefficients of the $p$-continued fraction are bounded above by $1/\sqrt{5}+\varepsilon_p$, where $\varepsilon_p\to 0$ as $p\to 0$. In light of Hurwitz's theorem, this upper bound is shar…
The problem of representing a given positive integer as a sum of four squares of integers has been widely concerned for a long time, and for a given positive odd $n$ one can find a representation by doing arithmetic in a maximal order of quaternion algebra once a pair of (positive) integers $x,y$ with $x^2+y^2\equiv-1\mod n$ is given. In this paper, we introduce a new method to find a representation of odd integer $w$ given $x,y$ satisfying the above requirement. This method can avoid the complicated non-commutative structure in quaternion algebra, which is similar to the one we use to obtain …
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）
Let $x$ be a irrational number in the unit interval and denote by its continued fraction expansion $[a_1(x), a_2(x), \cdots, a_n(x), \cdots]$. For any $n \geq 1$, write $T_n(x) = \max_{1 \leq k \leq n}\{a_k(x)\}$. We are interested in the Hausdorff dimension of the fractal set \[ E_ϕ= \left\{x \in (0,1): \lim_{n \to \infty} \frac{T_n(x)}{ϕ(n)} =1\right\}, \] where $ϕ$ is a positive function defined on $\mathbb{N}$ with $ϕ(n) \to \infty$ as $n \to \infty$. Some partial results have been obtained by Wu and Xu, Liao and Rams, and Ma. In the present paper, we further study this topic when $ϕ(n)$ t…
A classical result of Khinchin says that for almost all real numbers $α$, the geometric mean of the first $n$ digits $a_i(α)$ in the continued fraction expansion of $α$ converges to a number $K = 2.6854520\ldots$ (Khinchin's constant) as $n \to \infty$. On the other hand, for almost all $α$, the arithmetic mean of the first $n$ continued fraction digits $a_i(α)$ approaches infinity as $n \to \infty$. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the $1/k$-th powers of the $k$-th elementary symmetric means of $n$ numbers for $1 \leq k \leq n$. On…
Given a real number $α$ and a natural number $N$, the Sudler product is defined by $P_N(α) = \prod_{r=1}^{N} 2 \left\lvert \sin(π\left(rα\right))\right\rvert.$ Denoting by $F_n$ the $n$--th Fibonacci number and by $ϕ$ the Golden Ratio, we show that for $F_{n-1} \leq N < F_n$, we have $P_{F_{n-1}}(ϕ)\leq P_N(ϕ) \leq P_{F_{n}-1}(ϕ)$ and $\min_{N \geq 1} P_N(ϕ) = P_1(ϕ)$, thereby proving a conjecture of Grepstad, Kaltenböck and Neumüller. Furthermore, we find closed expressions for $\liminf_{N \to \infty} P_N(ϕ)$ and $\limsup_{N \to \infty} \frac{P_N(ϕ)}{N}$ whose numerical values can be approxim…
In this paper we study the relation between the function $J_{4_1,0}$, which arises from a quantum invariant of the figure-eight knot, and Sudler's trigonometric product. We find $J_{4_1,0}$ up to a constant factor along continued fraction convergents to a quadratic irrational, and we show that its asymptotics deviates from the universal limiting behavior that has been found by Bettin and Drappeau in the case of large partial quotients. We relate the value of $J_{4_1,0}$ to that of Sudler's trigonometric product, and establish asymptotic upper and lower bounds for such Sudler products in respon…
