The Lagrange spectrum $L$ is the set of finite values of the best approximation constants $k(α)=\limsup_{|p|,|q|\to \infty}|q(qα-p)|^{-1}$, where $α\in \mathbb{R}\setminus \mathbb{Q}$. It is a classical result that the pairs $(p,q)$ attaining these approximation constants arise from the convergents $(p_n,q_n)$ of the continued fraction of $α$. Consequently, $k(α)=\limsup_{n\to\infty}|q_n(q_nα-p_n)|^{-1}$. Moreira proved that the function $d(t)=HD(L\cap(-\infty,t))$ where $HD$ denotes Hausdorff dimension, is continuous.
Second Lagrange spectra are defined analogously to the classical Lagrange spectrum, but are associated with the problem of approximating an irrational number $α$ by rational numbers $\frac{p}{q}$ that are not convergents of its continued fraction expansion. Two natural definitions arise depending on whether rational multiples $(p,q)=(kp_n,kq_n),k\geq 2$ which represent the same rational numbers as convergents, are allowed or excluded. Based on this distinction, Moshchevitin introduced two second Lagrange spectra, denoted $L_2$ and $L_2^*$.
We prove that the function $d_2(t)=HD(L_2\cap (-\infty,t))$ is continuous, whereas $d_2^*(t)=HD(L_2^*\cap (-\infty,t))$ is discontinuous and assumes only the values 0 and 1.
We prove that, for every $k\ge 4$, the sets $M(k)$ and $L(k)$, which are Markov and Lagrange dynamical spectra related to conservative horseshoes and associated to continued fractions with coefficients bounded by $k$ coincide with the intersections of the classical Markov and Lagrange spectra with $(-\infty, \sqrt{k^2+4k}]$. We also observe that, despite the corresponding statement is also true for $k = 2$, it is false for $k = 3$.
We prove several results on (fractal) geometric properties of the classical Markov and Lagrange spectra. In particular, we prove that the Hausdorff dimensions of intersections of both spectra with half-lines always coincide, and may assume any real value in the interval $[0, 1]$.
On the graph of the dimension function of the Lagrange and Markov spectra
CNRS & École Polytechnique（法国国家科学研究中心与巴黎综合理工学院）；CNPq, FAPERJ, and INCTMAT project of J. Palis（巴西国家科学技术发展委员会、里约热内卢州研究资助基金会及J·帕里斯INCTMAT项目）；EPSRC grant EP/T001674/1（英国工程与物理科学研究委员会EP/T001674/1号基金）；School of Mathematical Sciences, Nankai University（南开大学数学科学学院）；IMPA（巴西纯粹与应用数学研究所）；Department of Mathematics, University of Surrey（萨里大学数学系）
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…
This text is a slightly expanded version of a survey article on certain aspects of low dimensional dynamics and number theory written after a kind invitation by the editors of the Notices of the American Mathematical Society.
We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excu…
We study Lagrange spectra of Veech translation surfaces, which are a generalization of the classical Lagrange spectrum. We show that any such Lagrange spectrum contains a Hall ray. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics in the corresponding Teichmüller disk and prove a formula which allows to express large values in the Lagrange spectrum as sums of Cantor sets.
We study Diophantine approximation in completions of functions fields over finite fields, and in particular in fields of formal Laurent series over finite fields. We introduce a Lagrange spectrum for the approximation by orbits of quadratic irrationals under the modular group. We give nonarchimedean analogs of various well known results in the real case: the closedness and boundedness of the Lagrange spectrum, the existence of a Hall ray, as well as computations of various Hurwitz constants. We use geometric methods of group actions on Bruhat-Tits trees.
The field of formal Laurent series is a natural analogue of the real numbers, and mathematicians have been translating well-known results about rational approximations to that setting. In the framework of power series over the rational numbers, we define and study the Lagrange spectrum, related to Diophantine approximation of irrationals, and the Markov spectrum, related to representation by indefinite binary quadratic forms. We compute both spectra explicitly, and show that they coincide and exhibit no gaps, contrary to what happens over the reals.
The Lagrange and Markov spectra are classical objects in Number Theory related to certain Diophantine approximation problems. Geometrically, they are the spectra of heights of geodesics in the modular surface. These objects were first studied by A. Markov in 1879, but, despite many efforts, the structure of the complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markov spectrum $M$ remained somewhat mysterious. In fact, it was shown by G. Freiman (in 1968 and 1973) and M. Flahive (in 1977) that $M\setminus L$ contains infinite \emph{countable} subsets near 3.11 and 3.29, and T. Cusic…
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.
In this note we will describe a simple and practical approach to get rigorous bounds on the Hausdorff dimension of limits sets for some one dimensional Markov iterated function schemes. The general problem has attracted considerable attention, but we are particularly concerned with the role of the value of the Hausdorff dimension in solving conjectures and problems in other areas red of mathematics. As our first application we confirm, and often strengthen, conjectures on the difference of the Lagrange and Markov spectra in Diophantine analysis, which appear in the work of Matheus and Moreira …
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。
Lagrange spectra have been defined for closed submanifolds of the moduli space of translation surfaces which are invariant under the action of SL(2,R). We consider the closed orbit generated by a specific covering of degree 7 of the standard torus, which is an element of the stratum H(2). We give an explicit formula for the values in the spectrum, in terms of a cocycle over the classical continued fraction. Differently from the classical case of the modular surface, where the lowest part of the Lagrange spectrum is discrete, we find an isolated minimum, and a set with a rich structure right ab…
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。
Let $L$ and $M$ denote the Lagrange and Markov spectra, respectively. It is known that $L\subset M$ and that $M\setminus L\neq\varnothing$. In this work, we exhibit new gaps of $L$ and $M$ using two methods. First, we derive such gaps by describing a new portion of $M\setminus L$ near to 3.938: this region (together with three other candidates) was found by investigating the pictures of $L$ recently produced by V. Delecroix and the last two authors with the aid of an algorithm explained in one of the appendices to this paper. As a by-product, we also get the largest known elements of $M\setmin…
It is well-known that Lagrange's four-square theorem, stating that every natural number may be written as the sum of four squares, may be proved using methods from the classical theory of modular forms and theta functions. We revisit this proof. In doing so, we concentrate on geometry and thereby avoid some of the tricky analysis that is often encountered. Guided by projective differential geometry we find a new route to Lagrange's theorem.
Freiman proved in 1968 that the Lagrange and Markov spectra do not coincide by exhibiting a countable infinite collection $\mathcal{F}$ of isolated points of the Markov spectrum which do not belong the Lagrange spectrum. In this paper, we describe the structure of the elements of the Markov spectrum in the largest interval $(c_{\infty}, C_{\infty})$ containing $\mathcal{F}$ and avoiding the Lagrange spectrum. In particular, we compute the smallest known element $f$ of $M\setminus L$, and we show that the Hausdorff dimension of the portion of the Markov spectrum between $c_{\infty}$ and $C_{\in…
