A Lagrange Theorem in dimension 2 is proved, for a particular two-dimensional algorithm, with a very natural geometrical definition. Dirichlet-type properties for the convergence of the algorithm are also proved. These properties procced from a geometrical quality of the algorithm. Some refercences are given to the works of various authors, in the domain of multidimensional continued fractions algorithms.
Let ${\bf F}_q$ be a finite field of order a positive power $q$ of a prime number. We study the nonarchimedean quadratic Lagrange spectrum defined by Parkkonen and Paulin by considering the approximation by elements of the orbit of a given quadratic power series in ${\bf F}_q((Y^{-1}))$, for the action by homographies and anti-homographies of ${\rm PGL}_2({\bf F}_q[Y])$ on ${\bf F}_q((Y^{-1})) \cup \{\infty\}$. While their approach used geometric methods of group actions on Bruhat--Tits trees, ours is based on the theory of continued fractions in power series fields.
We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give, for each of our algorithms, the complete characterization of elements having purely periodic expansions.
We prove an analogue of the Lagrange Inversion Theorem for Dirichlet series. The proof is based on studying properties of Dirichlet convolution polynomials, which are analogues of convolution polynomials introduced by Knuth in [4].
We give a new class of multidimensional $p$-adic continued fraction algorithms. We propose an algorithm in the class for which we can expect that multidimensional $p$-adic version of Lagrange's Theorem holds.
We establish a new transcendence criterion of $p$-adic continued fractions which are called Ruban continued fractions. By this result, we give explicit transcendental Ruban continued fractions with bounded $p$-adic absolute value of partial quotients. This is $p$-adic analogy of Baker's result. We also prove that $p$-adic analogy of Lagrange Theorem for Ruban continued fractions is not true.
The divided cell algorithm and the inhomogeneous Lagrange and Markoff spectra
Delone于1947年提出了计算二元二次型非齐次极小值的分胞算法，E. S. Barnes和H. P. F. 1950年代的Swinnerton-Dyer。我们展示了如何在过去五十年的进步，在符号计算和我们的理解均匀的光谱可以结合起来，使划分的细胞更有用的组织信息的非齐次逼近问题。我们分析的一个关键部分依赖于Jane Pitman的工作，他将分裂单元算法与常规连分数算法联系起来。特别地，与连分数的关系允许对同一问题的两个划分的单元进行比较，而不需要步进通过连接它们的划分的单元的链。
A well known theorem of Lagrange states that the simple continued fraction of a real number $α$ is periodic if and only if $α$ is a quadratic irrational. We examine non-periodic and non-simple continued fractions formed by two interlacing geometric series and show that in certain cases they converge to quadratic irrationalities. This phenomenon is connected with certain sequences of polynomials whose properties we examine further.
In this paper we study the geodesic continued fraction in the case of the Shimura curve coming from the $(2,3,7)$-triangle group. We construct a certain continued fraction expansion of real numbers using the so-called coding of the geodesics on the Shimura curve, and prove the Lagrange type periodicity theorem for the expansion which captures the fundamental relative units of quadratic extensions of $\mathbb{Q}(\cos(2π/7))$ with rank one relative unit groups. We also discuss the convergence of these continued fractions.
We consider continued fractions with partial quotients in the ring of integers of a quadratic number field $K$ and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the $β$-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number $β$, the $β$-continued fraction expansion of elements in $\mathbb Q(β)$ is either finite of eventually periodic. The same holds for $β$ being a square root of an integer. We also show that for certain 4 quadratic Per…
For a complex polynomial $D(t)$ of even degree, one may define the continued fraction of $\sqrt{D(t)}$. This was found relevant already by Abel in 1826, and later by Chebyshev, concerning integration of (hyperelliptic) differentials; they realized that, contrary to the classical case of square roots of positive integers treated by Lagrange and Galois, we do not always have pre-periodicity of the partial quotients. In this paper we shall prove that, however, a correct analogue of Lagrange's theorem still exists in full generality: pre-periodicity of the {\it degrees} of the partial quotients al…
Continued fractions have been introduced in the field of $p$--adic numbers $\mathbb{Q}_p$ by several authors. However, a standard definition is still missing since all the proposed algorithms are not able to replicate all the properties of continued fractions in $\mathbb{R}$. In particular, an analogue of the Lagrange's Theorem is not yet proved for any attempt of generalizing continued fractions in $\mathbb{Q}_p$. Thus, it is worth to study the definition of new algorithms for $p$--adic continued fractions. The main condition that a new method needs to fulfill is the convergence in $\mathbb Q…
In this paper, we present some generalizations of Lagrange's theorem in the classical theory of continued fractions motivated by the geometric interpretation of the classical theory in terms of closed geodesics on the modular curve. As a result, for an extension $F/F'$ of number fields with rank one relative unit group, we construct a geodesic multi-dimensional continued fraction algorithm to "expand" any basis of $F$ over $\mathbb{Q}$, and prove its periodicity. Furthermore, we show that the periods describe the relative unit group. By developing the above argument adelically, we also obtain …
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.
Our aim is to find a complex continued fraction algorithm finding all the best Diophantine approximations to a complex number. Using the sequence of minimal vectors in a two dimensional lattice over Gaussian integers, we obtain an algorithm defined on a submanifold of the space of unimodular two dimensional Gauss lattices. This submanifold is transverse to the diagonal flow. Thanks to the correspondence between minimal vectors and best Diophantine approximations, the algorithm finds all the best approximations to a complex number. A byproduct of the algorithm is the best constant for the compl…
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…
Periodicity and Period-Length Bounds for Browkin $p$-Adic Continued Fractions
Dipartimento di Matematica, Università di Trento（特伦托大学数学系）；Dipartimento di Scienze Matematiche ”Giuseppe Luigi Lagrange”, Politecnico di Torino（都灵理工学院朱塞佩·路易吉·拉格朗日数学科学系）
This paper is dedicated to the study of two famous subsets of the real line, namely Lagrange spectrum $L$ and Markov spectrum $M$. Our first result, Theorem 2.1, provides a rigorous estimate on the smallest value $t_1$ such that the portion of the Markov spectrum $(-\infty,t_1)\cap M$ has Hausdorff dimension $1$. Our second result, Theorem 3.1, gives a new upper bound on the Hausdorff dimension of the set difference $M\setminus L$. Our method combines new facts about the structure of the classical spectra together with finer estimates on the Hausdorff dimension of Gauss--Cantor sets of continu…
In this article we calculate the Hausdorff dimension of the set \begin{equation*} \mathcal{F}(Φ)=\left\{ x\in \lbrack 0,1):\begin{aligned}a_{n+1}(x)a_n(x) \geq Φ(n) \ {\rm for \ infinitely \ many \ } n\in \mathbb N \ {\rm and } \\ a_{n+1}(x)< Φ(n) \ {\rm for \ all \ sufficiently \ large \ } n\in \mathbb N \end{aligned}\right\} \end{equation*} where $Φ:\mathbb{N}\rightarrow (1,\infty)$ is any function with $\lim_{n\to \infty} Φ(n)=\infty.$ This in turn contributes to the metrical theory of continued fractions as well as gives insights about the set of Dirichlet non-improvable numbers.
We give an integrability condition on a function $ψ$ guaranteeing that for almost all (or almost no) $x\in\mathbb{R}$, the system $|qx-p|\leq ψ(t)$, $|q|<t$ is solvable in $p\in \mathbb{Z}$, $q\in \mathbb{Z}\smallsetminus \{0\}$ for sufficiently large $t$. Along the way, we characterize such $x$ in terms of the growth of their continued fraction entries, and we establish that Dirichlet's Approximation Theorem is sharp in a very strong sense. Higher-dimensional generalizations are discussed at the end of the paper.
Existence of a Hall ray in higher dimensional Lagrange spectra
Cheung Yau Mathematical Sciences Center, Tsinghua University（丘成桐数学科学中心，清华大学）；Department of Mathematics, University of Chicago（芝加哥大学数学系）；Qiuzhen College, Tsinghua University（清华大学邱成桐数学中心）
本文证明任意范数下二维 Lagrange 谱含 Hall 射线，结合 Kleinbock 结果，谱呈射线或射线前有稠密部分。
In 1848 Ch.~Hermite asked if there exists some way to write cubic irrationalities periodically. A little later in order to approach the problem C.G.J.~Jacobi and O.~Perron generalized the classical continued fraction algorithm to the three-dimensional case, this algorithm is called now the Jacobi-Perron algorithm. This algorithm is known to provide periodicity only for some cubic irrationalities. In this paper we introduce two new algorithms in the spirit of Jacobi-Perron algorithm: the heuristic algebraic periodicity detecting algorithm and the $\sin^2$-algorithm. The heuristic algebraic peri…
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…
We study a dynamical system that was originally defined by Romik in 2008 using an old theorem of Berggren concerning Pythagorean triples. Romik's system is closely related to the Farey map on the unit interval which generates an additive continued fraction algorithm. We explore some number theoretical properties of the Romik system. In particular, we prove an analogue of Lagrange's theorem in the case of the Romik system on the unit quarter circle, which states that a point possesses an eventually periodic digit expansion if and only if the point is defined over a real quadratic extension fiel…
Institute für diskrete Mathematik und Geometrie, Technische Universität Wien（维也纳工业大学离散数学与几何研究所）；Sydney Mathematical Research Institute, The University of Sydney（悉尼大学悉尼数学研究院）；Department of Mathematical and Physical Sciences, La Trobe University（拉筹伯大学数理科学系）
