Roswitha Hofer
The focus of this paper is on formal power series analogs of the golden ratio. We are interested in both their continued fractions expansions as well as their Laurent series expansions. Our approach studies the Hankel matrices that are determined using the coefficients of the Laurent series expansions.
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The classical continued fraction is generalized for studying the rational approximation problem on multi-formal Laurent series in this paper, the construction is called m-continued fraction. It is proved that the approximants of an m-continued fraction converge to a multi-formal Laurent series, and are best rational approximations to it; conversely for any multi-formal Laurent series an algorithm called m-CF transform is introduced to obtain its m-continued fraction expansions; moreover, strict m-continued fractions, which are m-continued fractions imposed with some additional conditions, and …
We study quadratic approximations for two families of hyperquadratic continued fractions in the field of Laurent series over a finite field. As the first application, we give the answer to a question of the second author concerning Diophantine exponents for algebraic Laurent series. As the second application, we determine the degrees of these families in particular case.
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…
Continued fractions of cubic Laurent series
本文为$\mathbb{Q}[[t^{-1}]]$中的Laurent级数族构造连分数展开式，应用超几何方法推导三次多项式实根有理逼近下界，并证明三次无理数存在可显式计算的闭形式连分数展开。
Fractal transference principle for continued fractions of Laurent series
本文基于Laurent级数连分数展开，建立分形转移原理，证明了在特定条件下，存在一个子集其连分数数字具有特定维度，并保持相对上密度。
Ramanujan recorded four reciprocity formulas for the Roger-Ramanujan continued fraction. Two reciprocity formulas each are also associated with the Ramanujan--Göllnitz--Gordon continued fraction and a level-13 analog of the Roger-Ramanujan continued fraction. We show that all eight reciprocity formulas are related to a pair of quadratic equations. The solution to the first equation generalizes the golden ratio and is used to set the value of a coefficient in the second equation; and the solution to the second equation gives a pair of values for a continued fraction. We relate the coefficients …
There is a theory of continued fractions for Laurent series in x^{-1} with coefficients in a field F. This theory bears a close analogy with classical continued fractions for real numbers with Laurent series playing the role of real numbers and the sum of the terms of non-negative degree in x playing the role of the integral part. In this paper we survey the Laurent series u, with coefficients in a finite extension of gf(2), that satisfy an irreducible equation of the form a_0(x)+ a_1(x)u+a_2(x)u^2 + a_3(x)u^3=0 with a_3 \ne 0 and where the a_i are polynomials of low degree in x with coefficie…
In this expository paper written to commemorate Fibonacci Day 2016, we discuss famous relations involving the Fibonacci sequence, the golden ratio, continued fractions and nested radicals, and show how these fit into a more general framework stemming from the quadratic formula.
For a monic polynomial $D(X)$ of even degree, express $\sqrt D$ as a Laurent series in $X^{-1}$; this yields a continued fraction expansion (similar to continued fractions of real numbers): \[\sqrt D=a_0+\dfrac{1}{a_1+\dfrac{1}{a_2+\dfrac{1}{\ddots}}},\quad a_i\text{ polynomials in }X.\] Such continued fractions were first considered by Abel in 1826, and later by Chebyshev. It turns out they are rarely periodic unless $D$ is defined over a finite field. Around 2001 van der Poorten studied non-periodic continued fractions of $\sqrt D$, with $D$ defined over the rationals, and simultaneously the…
An Engel series is a sum of the reciprocals of an increasing sequence of positive integers, which is such that each term is divisible by the previous one. Here we consider a particular class of Engel series, for which each term of the sequence is divisible by the square of the preceding one, and find an explicit expression for the continued fraction expansion of the sum of a generic series of this kind. As a special case, this includes certain series whose continued fraction expansion was found by Shallit. A family of examples generated by nonlinear recurrences with the Laurent property is con…
Hankel Matrix Decompositions and Digital Sequences over Function Fields
本文探究函数域上汉克尔矩阵与洛朗级数展开的关联，通过块$\boldsymbol{\textit{LB}\boldsymbol{\textit{U}}}$分解构造奥斯托夫斯基型序列，为利特伍德猜想反例搜索提供新路径，并证明Sobol'构造的分布性质。
We determine the classical approximation constants $w_{3}(ζ),w_{3}^{\ast}(ζ),λ_{3}(ζ)$ such as the uniform constants $\widehat{w}_{3}(ζ),\widehat{w}_{3}^{\ast}(ζ),\widehatλ_{3}(ζ)$ associated to real numbers $ζ$ whose continued fraction expansions are given by a Sturmian word. We more generally provide a description of the combined graph of the parametric successive minima functions defined by Schmidt and Summerer in dimension three for such Sturmian continued fractions. This both complements similar results due to Bugeaud and Laurent concerning the two-dimensional constants and generalizes a …
Euler gave recipes for converting alternating series of two types, I and II, into equivalent continued fractions, i.e., ones whose convergents equal the partial sums. A condition we prove for irrationality of a continued fraction then allows easy proofs that $e,\sin1$, and the primorial constant are irrational. Our main result is that, if a series of type II is equivalent to a simple continued fraction, then the sum is transcendental and its irrationality measure exceeds $2$. We construct all $\aleph_0^{\aleph_0}=\mathfrak{c}$ such series and recover the transcendence of the Davison--Shallit a…
It is possible to define a continued fraction expansion of elements in a function field of a curve by expanding as a Laurent series in a local parameter. Considering the square root of a polynomial $\sqrt{D(t)}$ leads to an interesting theory related to polynomial Pell equations. Unlike the classical Pell equation, the corresponding polynomial equation is not always solvable and its solvability is related to arithmetic conditions on the Jacobian (or generalized Jacobian) of the curve defined by $y^2=D(t)$. In this setting, it has been shown by Zannier in \cite{zannier} that the sequence of the…
We examine the polynomial analogues of McMullen's and Zaremba's conjectures on continued fractions with bounded partial quotients. It has already been proved by Blackburn that if the base field is infinite, then the polynomial analogue of Zaremba's conjecture holds; we will prove this again with a different method and examine some known results for finite base fields. Translating to the polynomial setting a result of Mercat, we will prove that the polynomial analogue of McMullen's conjecture holds over infinite algebraic extensions of finite fields and that, over finite fields, it would be a c…
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…
In this paper, we study Diophantine exponents $w_n$ and $w_n ^{*}$ for Laurent series over a finite field. Especially, we deal with the case $n=2$, that is, quadratic approximation. We first show that the range of the function $w_2-w_2 ^{*}$ is exactly the closed interval $[0,1]$. Next, we estimate an upper bound of the exponent $w_2$ of continued fractions with low complexity partial quotients.
We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction $[a_1,a_2,\ldots,a_n]$, we associate a snake graph $\mathcal{G}[a_1,a_2,\ldots,a_n]$ such that the continued fraction is the quotient of the number of perfect matchings of $\mathcal{G}[a_1,a_2,\ldots,a_n]$ and $\mathcal{G}[a_2,\ldots,a_n]$. We also show that snake graphs are in bijection with continued fractions. We then apply this conn…
Exercises in Iterational Asymptotics III
本文研究若干非线性递推，涵盖黄金比例、白银比例的简单连分数及关联 Abel 函数方程的参数化三次族。
Fractional Iterates and Oscillatory Convergence
本文研究了黄金均值和银均值的分数迭代，探讨了余弦和逻辑映射的振荡收敛特性。
On the Irrationality Exponents of Mahler Numbers
本文探讨了满足特定函数方程的 Mahler 数，利用形式 Laurent 级数的连分数开发了计算其无理指数的程序，并扩展了 Badziahin 在 $C(z)=0$ 条件下的研究。
Escape of mass of the Thue-Morse sequence
针对平面格点空间上概率测度序列完全质量逃逸的猜想，本文通过分析系数为Thue-Morse序列的F₂上洛朗级数的数墙显式对称性，否定了该猜想，推进了连分数部分商相关研究。
Hankel continued fractions and Hankel determinants for $q$-deformed metallic numbers
本文研究了$q$-变形金属数的Hankel连续分数和行列式性质，证明其序列具有周期性和递推关系，并验证了相关猜想。
Quantum Kronecker fractions
本文研究收敛半径为1的有理数（称为克罗内克分数），证明其连分数展开为回文，描述素分母情况，发现无限族及分母小于5000的所有此类分数，并评述无理数情况及与辫子、有理结链的关系。
On some Diophantine and Ergodic properties of the Schneider map over function fields
该论文研究函数域上Schneider映射定义的连分数，证明其Laurent级数域元素的连分数展开存在唯一，确立相关丢番图逼近与遍历性质，确定最难逼近元素集的Hausdorff维数，为函数域数论与动力系统提供新结果。
Let $\wh K$ be the field of formal Laurent series in $X^{-1}$ over the finite field $k$, and let $A$ be the ring of polynomials in $X$ over $k$. One of the main results of the paper is to give a particularly nice coding of the geodesic flow on the quotient of the Bruhat-Tits tree $\TT$ of ${\rm PGL}\_2(\wh K)$ by ${\rm PGL}\_2(A)$, by using the continued fraction expansion of the endpoints of the geodesic lines in $\TT$ (the space of ends of $\TT$ identifies with $\PP\_1(\wh K)$). This allows in particular to prove in a dynamical way the invariance of the Haar measure by the Artin map.
Higher Dimer Covers on Snake Graphs
本文研究蛇图的m-二聚体覆盖计数问题，证明其数量对应(m+1)阶矩阵乘积的左上角元素，推广了连分数的组合解释，并为三次无理数的埃尔米特问题提供新方向。
Regularity properties of the $α$-Wilton functions
本文研究$α$-连分数相关Wilton函数的正则性，证明其在$α\in[1-g,g]$上为BMO且该区间最优，并利用匹配性质给出边界邻域内非BMO的反例。
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