Oleg Karpenkov
The problem of investigation of the simplest n-dimensional continued fraction in the sense of Klein for n>2 was posed by V.Arnold. The answer for the case of n=2 can be found in the works of E.Korkina and G.Lachaud. In present work we study the case of n=3.
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We consider the geometric generalization of ordinary continued fraction to the multidimensional case introduced by F. Klein in 1895. A multidimensional periodic continued fraction is the union of sails with some special group acting freely on these sails. This group transposes the faces. In this article, we present a method of constructing "approximate" fundamental domains of algebraic multidimensional continued fractions and an algorithm testing whether this domain is indeed fundamental or not. We give some polynomial estimates on number of the operations for the algorithm. In conclusion we p…
The Lagrange theorem on continued fractions states that a number is a quadratic surd if and only if its continued fraction expansion is eventually periodic. The current paper is devoted to a multidimensional generalization of this fact. As a multidimensional analog of continued fractions so called Klein polyhedra are considered: given an irrational lattice in an n-dimensional Euclidean space, its Klein polyhedron is defined as the convex hull of nonzero lattice points with nonnegative coordinates.
Complex continued fractions, Kleinian and extremal theory for cusp excursions
本文研究了五个复二次整数欧氏环上的复连分数算法，证明其最大数字服从Fréchet分布，并据此求出相关Bianchi轨形上尖点远足的极限分布。
In this work we prove a criterion for an algebraic continued fraction to have a proper palindromic symmetry in dimension $4$. As a multidimensional generalization of continued fractions, we consider Klein polyhedra.
We show, for each $q$-continued fraction $G(q)$ in a certain class of continued fractions, that there is an uncountable set of points on the unit circle at which $G(q)$ diverges in the general sense. This class includes the Rogers-Ramanujan continued fraction and the three Ramanujan-Selberg continued fraction. We discuss the implications of our theorems for the general convergence of other $q$-continued fractions, for example the Göllnitz-Gordon continued fraction, on the unit circle.
If the odd and even parts of a continued fraction converge to different values, the continued fraction may or may not converge in the general sense. We prove a theorem which settles the question of general convergence for a wide class of such continued fractions. We apply this theorem to two general classes of $q$ continued fraction to show, that if $G(q)$ is one of these continued fractions and $|q|>1$, then either $G(q)$ converges or does not converge in the general sense. We also show that if the odd and even parts of the continued fraction $K_{n=1}^{\infty}a_{n}/1$ converge to different va…
This work is devoted to the proof of the statement about the existence of palindromic continued fractions in an arbitrary dimension. In addition, it is proved the criterion that an algebraic continued fraction has proper cyclic palindromic symmetry in the case $n = 4$. Klein polyhedra are considered as a multidimensional generalization of continued fractions.
In this paper we prove in detail a criterion for an algebraic continued fraction to have a proper palindromic symmetry in dimension $4$. We also present a new proof of the criterion for an algebraic continued fraction to have a proper cyclic palindromic symmetry in dimension $4$. As a multidimensional generalization of continued fractions, we consider Klein polyhedra.
This paper is a short survey of the recent results on examples of periodic two-dimensional continued fractions (in Klein's model). In the last part of this paper we formulate some questions, problems and conjectures on geometrical properties concerning to this subject.
For a given sequence of positive integers we make an explicit construction of a reduced hyperbolic operator in SL(2,z) with the sequence as a period of a geometric continued fraction in the sense of Klein. Further we experimentally study an algorithm to construct a period for an arbitrary operator of SL(2,z) (the Gauss Reduction Theory).
In this paper we describe the group of symmetries of a two-dimensional continued fraction. As a multidimensional generalization of continued fractions we consider Klein polyhedra. We distinguish two types of symmetries: the Dirichlet-type ones, that correspond to the multiplication by units of the corresponding number field, and so called palindromic ones. The main result of the paper is a criterion for a two-dimensional continued fraction to have a palindromic symmetry. This criterion is analogous to the criterion for the period of a quadratic irrationality to be symmetric.
The study of combinatorial properties of mathematical objects is a very important research field and continued fractions have been deeply studied in this sense. However, multidimensional continued fractions, which are a generalization arising from an algorithm due to Jacobi, have been poorly investigated in this sense, up to now. In this paper, we propose a combinatorial interpretation of the convergents of multidimensional continued fractions in terms of counting some particular tilings, generalizing some results that hold for classical continued fractions.
It has been known that the Hosoya index of caterpillar graph can be calculated as the numerator of the simple continued fraction. Recently, the author \cite{Komatsu2020} introduces a more general graph called caterpillar-bond graph and shows that its Hosoya index can be calculated as the numerator of the general continued fraction. In this paper, we show how the Hosoya index of the graph with non-uniform ring structure can be calculated from the negative continued fraction. We also give the relation between some radial graphs and multidimensional continued fractions in the sense of the Hosoya …
In this paper we study the three-dimensional analogue of the relation between the irrationality exponent of a real number and the growth of its regular continued fraction partial quotients. As a multidimensional generalisation of continued fractions, we consider Klein polyhedra.
Geometry of multidimensional Farey summation algorithm and frieze patterns
该研究提出高维减法连分数算法的新几何方法，引入推广Klein多面体的Farey多面体及其帆，用棱柱图分类，得到多维Conway-Coxeter带状图案并证明其满足推广托勒密关系。
Continued Fractions and the 4-Color Theorem
本文通过连分数的几何解释，研究了具有特定度序列的球面三角剖分的四色着色问题，并提出一个猜想性的等周不等式。
The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approximation coefficients (in the sense of Diophantine approximation by continued fraction convergents). We also obtain metrical results for large blocks of ``bad'' approximations.
We study a family of generalized continued fractions, which are defined by a pair of substitution sequences in a finite alphabet. We prove that they are stammering sequences, in the sense of Adamczewski and Bugeaud. We also prove that this family consists of transcendental numbers which are not Liouvillian. We explore the partial quotients of their regular continued fraction expansions, arriving at no conclusion concerning their boundedness.
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.
In this paper we study the forbidden values of the conductor $q$ of the $L$-functions of degree 2 in the extended Selberg class by a novel technique, linking the problem to certain continued fractions and to their weight $w_q$. Our basic result states that if an $L$ function with conductor $q$ exists, then the weight $w_q$ is unique in a suitable sense. From this we deduce several results, both of theoretical and computational nature.
Asymptotic statistics for finite continued fractions with restricted digits
本文研究了有限继续分数在受限数字下的渐近统计特性，通过分析特定分形集的ε-增厚大小，对Zaremba猜想进行了平均意义下的评论，并讨论了复继续分数在虚二次域上的推广。
In the theory of continued fractions, Zaremba's conjecture states that there is a positive integer $M$ such that each integer is the denominator of a convergent of an ordinary continued fraction with partial quotients bounded by $M$. In this paper, to each such $M$ we associate a regular sequence---in the sense of Allouche and Shallit---and establish various properties and results concerning the generating function of the regular sequence. In particular, we determine the minimal algebraic relation concerning the generating function and its Mahler iterates.
There exists and is unique up to multiplication by a constant function a form of the highest dimension on the manifold of n-dimensional continued fractions in the sense of Klein, such that the form is invariant under the natural action of the group of projective transformations PGL(n+1). A measure corresponding to the integral of such form is called a Mobius measure. In the present paper we deduce an explicit formulae to calculate invariant forms in special coordinates. These formulae allow to give answers to some statistical questions of theory of multidimensional continued fractions. As an e…
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。
Generalised Gauss--Kuzmin Distribution for Klein sails in $\mathbb R^3$
本文将经典高斯-库兹明分布推广至三维空间，证明一般克莱因帆局部面结构的等分布，解决Karpenkov公开问题，通过构造SL₃(ℝ)/SL₃(ℤ)对角流的有限截面测度推导极限面统计量。
A finiteness condition for complex continued fraction algorithms
该研究针对复连分数算法的简单性刻画需求，证明参数为有理数对的复α-Hurwitz算法满足有限范围条件，等价于存在相关有限划分，为相关学者的近期问题提供了部分解答。
The continued fraction expansion of an irrational number $α$ is eventually periodic if and only if $α$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real numbers of degree at least three. Because of some numerical evidence and a belief that these numbers behave like most numbers in this respect, it is often conjectured that their partial quotients form an unbounded sequence. More modestly, we may expect that if the sequence of partial quotients of an irrational number $α$ is, in some sense, "simple", then $α$ is either quad…
A Continued Fractions Theory for the completion of the Puiseux field
本文建立了 Puiseux 级数域拓扑完备化上的连分数理论，证明任意元素具有唯一连分数展开且给出最佳有理丢番图逼近，并将其与 Berkovich 空间上算术子群作用及 IV 型点联系起来。
A Trott number is a number $x\in(0,1)$ whose continued fraction expansion is equal to its base $b$ expansion for a given base $b$, in the following sense: If $x=[0;a_1,a_2,\dots]$, then $x=(0.\hat{a}_1\hat{a}_2\dots)_b$, where $\hat{a}_i$ is the string of digits resulting from writing $a_i$ in base $b$. In this paper we characterize the set of bases for which Trott numbers exist, and show that for these bases, the set $T_b$ of Trott numbers is a complete $G_δ$ set. We prove moreover that the union $T:=\bigcup_{b\geq 2} T_b$ is nowhere dense and has Hausdorff dimension less than one. Finally, w…
Let $β> 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $β$-expansion of $x$ ($n \in \mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\log2\logβ)/π^2$ almost everywhere in the sense of Lebesgue measure. In this paper, we improve this result by proving that the Lebesgue measure of the set of $x \in [0,1)$ for which $k_n(x)/n$ deviates away from $(6\log2\logβ)/π^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes …
The present paper is in a sense a continuation of \cite{PLS}, it relies on the notation and some results. The problem tackled in both papers is the nature of the continued fraction expansion of $\sqrt[3]{2}$: are the partial quotients bounded or not. Numerical experiments suggest an even stronger result on the lines of Kuzmin statistics. Here we apply different sets of bases for the vector space $V$, where the adjunction ring $\ZZ\lbrack \sqrt[3]{2} \rbrack$ lives. And as a result we get a criterion for continued fraction convergents in terms of the coefficient vector from a lattice.
An Elementary Characterization of the Gauss--Kuzmin Measure in the Theory of Continued Fractions
本文证明连分数理论中 Gauss--Kuzmin 测度可由其逆序对称性唯一刻画，并探讨了串排列的非平凡对称性存在性及其渐近普遍性。
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