Lagrange Spectrum Research Articles

Fractal dimensions of the Markov and Lagrange spectra near $3$

The Lagrange spectrum \mathcal{L} and Markov spectrum \mathcal{M} are subsets of the real line with complicated fractal properties that appear naturally in the study of Diophantine approximations. It is known that the Hausdorff dimensions of the intersections of these sets with any half-line coincide, that is, \dim_{H}(\mathcal{L}\cap (-\infty, t)) = \dim_{H}(\mathcal{M}\cap (-\infty, t))\equalscolon d(t) for every t \geq 0 . It is also known that d(3)=0 and d(3+\varepsilon)>0 for every \varepsilon>0 .We show that, for sufficiently small values of \varepsilon > 0 , one has the approximation d(3+\varepsilon) = 2\cdot\frac{W(e^{c_0}|\log \varepsilon|)}{|\log \varepsilon|}+\mathrm{O}(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^{2}}) , where W denotes the Lambert function (the inverse of f(x)=xe^{x} ) and c_{0}=-\log\log((3+\sqrt{5})/2) \approx 0.0383 . We also show that this result is optimal for the approximation of d(3+\varepsilon) by “reasonable” functions, in the sense that, if F(t) is a C^{2} function such that d(3+\varepsilon) = F(\varepsilon) + \mathrm{o}(\frac{\log |\log \varepsilon|}{|\log \varepsilon|^{2}}) , then its second derivative F''(t) changes sign infinitely many times as t approaches 0 .

Markoff–Lagrange spectrum of one‐sided shifts

AbstractFor the Lagrange spectrum and other applications, we determine the smallest accumulation point of binary sequences that are maximal in their shift orbits. This problem is trivial for the lexicographic order, and its solution is the fixed point of a substitution for the alternating lexicographic order. For orders defined by cylinders, we show that the solutions are ‐adic sequences, where is a certain infinite set of substitutions that contains Sturmian morphisms. We also consider a similar problem for symmetric ternary shifts, which is applicable to the multiplicative version of the Markoff–Lagrange spectrum.

Intrinsic Diophantine approximation on circles and spheres

AbstractWe study Lagrange spectra arising from intrinsic Diophantine approximation of circles and spheres. More precisely, we consider three circles embedded in or and three spheres embedded in or . We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of and . Thanks to prior work of Asmus L. Schmidt on the spectra of and , we obtain as a corollary, for each of the six spectra, the smallest accumulation point and the initial discrete part leading up to it completely.

Corrigendum and addendum to Appendix A of “Fractal geometry of the complement of Lagrange spectrum in Markov spectrum”

This erratum corrects the statement and proof of Corollary A.4 of [Comment. Math. Helv. 95, 593–633 (2020)].

Pinwheels as Lagrangian barriers

The complex projective plane [Formula: see text] contains certain Lagrangian CW-complexes called pinwheels, which have interesting rigidity properties related to solutions of the Markov equation, see for example [J. Evans and I. Smith, Markov numbers and Lagrangian cell complexes in the complex projective plane, Geom. Topol. 22 (2018) 1143–1180]. We compute the Gromov width of the complement of pinwheels and show that it is strictly smaller than the Gromov width of [Formula: see text], meaning that pinwheels are Lagrangian barriers in the sense of [P. Biran, Lagrangian barriers and symplectic embeddings, Geom. Funct. Anal. 11 (2001) 407–464]. The accumulation points of the set of these Gromov widths are in a simple bijection with the Lagrange spectrum below [Formula: see text].

Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum

Let ℒ(S 1 ) be the Lagrange spectrum arising from intrinsic Diophantine approximation on the unit circle S 1 by its rational points. We give a complete description of the structure of ℒ(S 1 ) below its smallest accumulation point. To this end, we use digit expansions of points on S 1 , which were originally introduced by Romik in 2008 as an analogue of simple continued fraction of a real number. We prove that the smallest accumulation point of ℒ(S 1 ) is 2. Also we characterize the points on S 1 whose Lagrange numbers are less than 2 in terms of Romik’s digit expansions. Our theorem is the analogue of the celebrated theorem of Markoff on badly approximable real numbers.

Diophantine approximation on conics

Given a conic C \mathcal {C} over Q \mathbb {Q} , it is natural to ask what real points on C \mathcal {C} are most difficult to approximate by rational points of low height. For the analogous problem on the real line (for which the least approximable number is the golden ratio, by Hurwitz’s theorem), the approximabilities comprise the classically studied Lagrange and Markoff spectra, but work by Cha–Kim [Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum, arXiv: 1903.02882, 2021] and Cha–Chapman–Gelb–Weiss [Monatsh. Math. 197 (2022), pp. 1–55] shows that the spectra of conics can vary. We provide notions of approximability, Lagrange spectrum, and Markoff spectrum valid for a general C \mathcal {C} and prove that their behavior is exhausted by the special family of conics C n : X Z = n Y 2 \mathcal {C}_n : XZ = nY^2 , which has symmetry by the modular group Γ 0 ( n ) \Gamma _0(n) and whose Markoff spectrum was studied in a different guise by A. Schmidt and Vulakh [Trans. Amer. Math. Soc. 352 (2000), pp. 4067–4094]. The proof proceeds by using the Gross-Lucianovic bijection to relate a conic to a quaternionic subring of Mat 2 × 2 ⁡ ( Z ) \operatorname {Mat}^{2\times 2}(\mathbb {Z}) and classifying invariant lattices in its 2 2 -dimensional representation.

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups

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 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 [Comment. Math. Helv. 95 (2020), pp. 593–633]. As a second application we (re-)validate and improve estimates connected with the Zaremba conjecture in number theory, used in the work of Bourgain–Kontorovich [Ann. of Math. (2) 180 (2014), pp. 137–196], Huang [An improvement to Zaremba’s conjecture, ProQuest LLC, Ann Arbor, MI, 2015] and Kan [Mat. Sb. 210 (2019), pp. 75–130]. As a third more geometric application, we rigorously bound the bottom of the spectrum of the Laplacian for infinite area surfaces, as illustrated by an example studied by McMullen [Amer. J. Math. 120 (1998), pp. 691-721]. In all approaches to estimating the dimension of limit sets there are questions about the efficiency of the algorithm, the computational effort required and the rigour of the bounds. The approach we use has the virtues of being simple and efficient and we present it in this paper in a way that is straightforward to implement. These estimates apparently cannot be obtained by other known methods.

Dynamical characterization of initial segments of the Markov and Lagrange spectra

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 $$k=3$$ .

Hausdorff dimension of Gauss–Cantor sets and two applications to classical Lagrange and Markov spectra

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 t1 such that the portion of the Markov spectrum (−∞,t1)∩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∖L. In addition, we also give a plot of the dimension function, which hasn't appeared previously in the literature to our knowledge.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 continued fraction expansions whose entries satisfy appropriate restrictions.

Diophantine Approximation, Lagrange and Markov Spectra, and Dynamical Cantor Sets

Diophantine Approximation, Lagrange and Markov Spectra, and Dynamical Cantor Sets

Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II

AbstractLet $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface N, and let $\Lambda _0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation g of $g_0$ and a smooth real-valued function f on the unit tangent bundle to N with respect to g, let $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$ ) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively highest) values of f along the geodesics in the hyperbolic continuation $\Lambda $ of $\Lambda _0$ . We prove that for generic choices of g and f, the Hausdorff dimensions of the sets $L_{g,\Lambda , f}\cap (-\infty , t)$ vary continuously with $t\in \mathbb {R}$ and, moreover, $M_{g,\Lambda , f}\cap (-\infty , t)$ has the same Hausdorff dimension as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$ .

Markov and Lagrange spectra for Laurent series in $1/T$ with rational coefficients

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

Phase transitions on the Markov and Lagrange dynamical spectra

The Markov and Lagrange dynamical spectra were introduced by Moreira and share several geometric and topological aspects with the classical ones. However, some features of generic dynamical spectra associated to hyperbolic sets can be proved in the dynamical case and we do not know if they are true in classical case.They can be a good source of natural conjectures about the classical spectra: it is natural to conjecture that some properties which hold for generic dynamical spectra associated to hyperbolic maps also hold for the classical Markov and Lagrange spectra.In this paper, we show that, for generic dynamical spectra associated to horseshoes, there are transition points a and a˜ in the Markov and Lagrange spectra respectively, such that for any δ>0, the intersection of the Markov spectrum with (−∞,a−δ) has Hausdorff dimension smaller than one, while the intersection of the Markov spectrum with (a,a+δ) has non-empty interior. Similarly, the intersection of the Lagrange spectrum with (−∞,a˜−δ) has Hausdorff dimension smaller than one, while the intersection of the Lagrange spectrum with (a˜,a˜+δ) has non-empty interior. We give an open set of examples where a≠a˜ and we prove that, in the conservative case, generically, a=a˜ and, for any δ>0, the intersection of the Lagrange spectrum with (a−δ,a) has Hausdorff dimension one.

Fractal geometry of the complement of Lagrange spectrum in Markov spectrum

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 countable subsets near 3.11 and 3.29, and T. Cusick conjectured in 1975 that all elements of M\setminus L were < \sqrt{12} = 3.46\ldots , and this was the status quo of our knowledge of M\setminus L until 2017. In this article, we show the following two results. First, we prove that M\setminus L is richer than it was previously thought because it contains a Cantor set of Hausdorff dimension larger than 1/2 near 3.7: in particular, this solves (negatively) Cusick's conjecture mentioned above. Secondly, we show that M\setminus L is not very thick: its Hausdorff dimension is strictly smaller than one.

The beginning of the Lagrange spectrum of certain origamis of genus two

The beginning of the Lagrange spectrum of certain origamis of genus two

Eigentime identity of the weighted (m,n)-flower networks

The eigentime identity for random walks on the weighted networks is the expected time for a walker going from a node to another node. Eigentime identity can be studied by the sum of reciprocals of all nonzero Laplacian eigenvalues on the weighted networks. In this paper, we study the weighted [Formula: see text]-flower networks with the weight factor [Formula: see text]. We divide the set of the nonzero Laplacian eigenvalues into three subsets according to the obtained characteristic polynomial. Then we obtain the analytic expression of the eigentime identity [Formula: see text] of the weighted [Formula: see text]-flower networks by using the characteristic polynomial of Laplacian and recurrent structure of Markov spectrum. We take [Formula: see text], [Formula: see text] as example, and show that the leading term of the eigentime identity on the weighted [Formula: see text]-flower networks obey superlinearly, linearly with the network size.

M∖L is Not Closed

Abstract We show that $1+3/\sqrt{2}$ is a point of the Lagrange spectrum $L$ that is accumulated by a sequence of elements of the complement $M\!\setminus\! L$ of the Lagrange spectrum in the Markov spectrum $M$. In particular, $M\!\setminus\! L$ is not a closed subset of $\mathbb{R}$, so that a question by T. Bousch has a negative answer.

Applications of Markov spectra for the weighted partition network by the substitution rule

The weighted self-similar network is introduced in an iterative way. In order to understand the topological properties of the self-similar network, we have done a lot of research in this field. Firstly, according to the symmetry feature of the self-similar network, we deduce the recursive relationship of its eigenvalues at two successive generations of the transition-weighted matrix. Then, we obtain eigenvalues of the Laplacian matrix from these two successive generations. Finally, we calculate an accurate expression for the eigentime identity and Kirchhoff index from the spectrum of the Laplacian matrix.

Generalised Markov numbers

In this paper we introduce generalised Markov numbers and extend the classical Markov theory for the discrete Markov spectrum to the case of generalised Markov numbers. In particular we show recursive properties for these numbers and find corresponding values in the Markov spectrum. Further we construct a counterexample to the generalised Markov uniqueness conjecture. The proposed generalisation is based on geometry of numbers. It substantively uses lattice trigonometry and geometric theory of continued numbers.