arXiv subjects

Seonhee Lim

Publications and source records attributed to Seonhee Lim.

At least 19 recordsLinked to original sources

arXiv2026

Extreme value theorem for geodesic flow on the quotient of the theta group

We establish an extreme value theorem for the geodesic flow on the hyperbolic surface $\Theta\backslash\mathbb{H}^2$ associated with the theta group $\Theta$. To capture excursions into both cusps of this surface, we introduce a generalized continued fraction algorithm obtained by splicing the even and odd-odd continued fraction maps into a single dynamical system. We prove that the natural extension of this map is isomorphic to the first return map of the geodesic flow on a suitable cross section. Using spectral properties of the associated transfer operator, we derive a Galambos-type extreme value law for the digits of the spliced continued fraction. This symbolic result is then translated into a geometric extreme value theorem describing maximal cusp excursions of geodesics on $\Theta\backslash\mathbb{H}^2$.

math.DS↗

arXiv2024

Singular systems of linear forms over global function fields

In this paper, we consider singular systems of linear forms over global function fields of class number one and give an upper bound for the Hausdorff dimension of the set of singular systems of linear forms by constructing an appropriate Margulis height function on the space of lattices over global function fields.

math.DS↗

arXiv2024

Euclidean algorithms are Gaussian over imaginary quadratic fields

The distributional analysis of Euclidean algorithms was carried out by Baladi and Vall\'{e}e. They showed the asymptotic normality of the number of division steps and associated costs in the Euclidean algorithm as a random variable on the set of rational numbers with bounded denominator based on the transfer operator methods. We extend their result to the Euclidean algorithm over appropriate imaginary quadratic fields by studying dynamics of the nearest integer complex continued fraction map, which is piecewise analytic and expanding but not a full branch map. By observing a finite Markov partition with a regular CW-structure, which enables us to associate the transfer operator acting on a direct sum of spaces of $C^1$-functions, we obtain the limit Gaussian distribution as well as residual equidistribution.

math.DS↗

arXiv2023

$\operatorname{PGL}_{2}(\mathbb{Q}_{p})$-orbit closures on a $p$-adic homogeneous space of infinite volume

Let $\mathbb{K}$ be an unramified quadratic extension of $\mathbb{Q}_{p}$ for a fixed $p>2$. Projective general linear groups $G=\operatorname{PGL}_{2}(\mathbb{K})$ and $H=\operatorname{PGL}_{2}(\mathbb{Q}_{p})$ act transitively on Bruhat-Tits trees $T_G$ and $T_H$, respectively. We identify $G/H$ with the set of $H$-subtrees $G.T_{H}$. Let $\Gamma$ be a Schottky subgroup such that $\Gamma\backslash T_{G}$ is infinite volume and has an additional condition named high-branchedness, and let $\Lambda$ be its limit set. We classify $\Gamma$-orbits in $G/H$. Let $C=g_{C}H\in G/H$. As a generalization of Ratner's theorem, if $\Gamma\backslash g_{C}.T_{H}$ meets the convex core of $\Gamma\backslash T_{G}$, then the $\Gamma$-orbit of $C$ is either dense or closed in $ {\cal{C}}_{\Lambda}=\{g H: \partial(g.T_{H})\cap\Lambda\neq\varnothing\}$.

math.GR↗

arXiv2021

On Hausdorff dimension in inhomogeneous Diophantine approximation over global function fields

In this paper, we study inhomogeneous Diophantine approximation over the completion $K_v$ of a global function field $K$ (over a finite field) for a discrete valuation $v$, with affine algebra $R_v$. We obtain an effective upper bound for the Hausdorff dimension of the set \[ \mathbf{Bad}_A(\epsilon)=\left\{\boldsymbol{\theta}\in K_v^{\,m} : \liminf_{(\mathbf{p},\mathbf{q})\in R_v^{\,m} \times R_v^{\,n}, \|\mathbf{q}\|\to \infty} \|\mathbf{q}\|^n \|A\mathbf{q}-\boldsymbol{\theta}-\mathbf{p}\|^m \geq \epsilon \right\}, \] of $\epsilon$-badly approximable targets $\boldsymbol{\theta}\in K_v^{\,m}$ for a fixed matrix $A\in\mathscr{M}_{m,n}(K_v)$, using an effective version of entropy rigidity in homogeneous dynamics for an appropriate diagonal action on the space of $R_v$-grids. We further characterize matrices $A$ for which $\mathbf{Bad}_A(\epsilon)$ has full Hausdorff dimension for some $\epsilon>0$ by a Diophantine condition of singularity on average. Our methods also work for the approximation using weighted ultrametric distances.

math.NT↗

arXiv2021

Dimension estimates for badly approximable affine forms

For given $\epsilon>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $\epsilon$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq \epsilon,$$ where $\langle \cdot \rangle$ denotes the distance from the nearest integral point. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $\epsilon$-badly approximable matrices for fixed target $b$ and the set of $\epsilon$-badly approximable targets for fixed matrix $A$. Moreover, we give an equivalent Diophantine condition of $A$ for which the set of $\epsilon$-badly approximable targets for fixed $A$ has full Hausdorff dimension for some $\epsilon>0$. The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the $A$-fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms.

math.DS↗

arXiv2021

Asymptotic distribution for pairs of linear and quadratic forms at integral vectors

We study the joint distribution of values of a pair consisting of a quadratic form $q$ and a linear form $\mathbf l$ over the set of integral vectors, a problem initiated by Dani-Margulis (1989). In the spirit of the celebrated theorem of Eskin, Margulis and Mozes on the quantitative version of the Oppenheim conjecture, we show that if $n \ge 5$ then under the assumptions that for every $(\alpha, \beta ) \in \mathbb R^2 \setminus \{ (0,0) \}$, the form $\alpha q + \beta \mathbf l^2$ is irrational and that the signature of the restriction of $q$ to the kernel of $\mathbf l$ is $(p, n-1-p)$, where $3\le p \le n-2$, the number of vectors $v \in \mathbb Z^n$ for which $\|v\| < T$, $a < q(v) < b$ and $c< \mathbf l(v) < d$ is asymptotically $$ C(q, \mathbf l)(d-c)(b-a)T^{n-3} , $$ as $T \to \infty$, where $C(q, \mathbf l)$ only depends on $q$ and $\mathbf l$. The density of the set of joint values of $(q, \mathbf l)$ under the same assumptions is shown by Gorodnik (2004).

math.DS↗

arXiv2020

Martin boundary of Brownian motion on Gromov hyperbolic metric graphs

Let $\widetilde{X}$ be a locally finite complete Gromov hyperbolic metric graph with the geometric boundary consisting of infinitely many points. Suppose that there is a discrete subgroup of the isometry group $Iso(\widetilde{X})$ acting geometrically on $\widetilde{X}$. The $λ$-Martin boundary is the boundary of the image of an embedding from $\widetilde{X}$ to the space of $λ$-superharmonic functions. We show that the $λ$-Martin boundary coincides with the geometric boundary for any $λ\in [0, λ_0],$ in particular at the bottom of the spectrum $λ_0$.

math.DS↗

arXiv2020

Local Limit Theorem in negative curvature

Consider the heat kernel $p(t,x,y)$ on the universal cover $X$ of a Riemannian manifold $M$ of negative curvature. We show the local limit theorem for $p$ : $$\lim_{t \to \infty} t^{3/2}e^{λ_0 t} p(t,x,y)=C(x,y),$$ where $λ_0$ is the bottom of the spectrum of the geometric Laplacian and $C(x,y)$ is a positive function which depends on $x, y \in X$. We also show that the $λ_0$-Martin boundary of $X$ is equal to its topological boundary. The Martin decomposition of $C(x,y)$ gives a family of measures $\{μ^{λ_0}_x \}$ on $\partial \widetilde{M}$. We show that $\{μ^{λ_0}_x \}$ is the unique family minimizing the energy or the Rayleigh quotient of Mohsen. We use the uniform Harnack inequality on the boundary $\partial X$ and the uniform three-mixing of the geodesic flow on the unit tangent bundle $SM$ for suitable Gibbs-Margulis measures.

math.DS↗

arXiv2019

Dimension bound for doubly badly approximable affine forms

We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $ε$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the Hausdorff dimension of the set $Bad_A(ε)$ of $ε$-badly approximable target $b$ is not full, and that for real numbers $α$, $\dim_H Bad_α(ε)=1$ if and only if $α$ is singular on average. We show that if $\dim_H Bad_A(ε)=m$, then $A$ is singular on average.

math.DS↗

arXiv2019

Limiting distribution of geodesics in a geometrically finite quotients of regular trees

In this article, we prove an extreme value theorem on the limit distribution of geodesics in a geometrically finite quotient of $Γ\backslash\mathcal{T}$ a locally finite tree. Main examples of such graphs are quotients of a Bruhat-Tits tree $\mathcal{T}$ by non-cocompact discrete subgroups $Γ$ of $PGL(2,\mathbf{K})$ of a positive characteristic local field $\mathbf{K}$. We investigate, for a given time $T$, the measure of the set of $Γ$-equivalent geodesic classes which stay up to time $T$ the region of distance $d$ at most $N$ depending on $T$ from a fixed compact subset $D$ of $Γ\backslash\mathcal{T}$. Namely, for Bowen-Margulis measure $μ$ on the space $Γ\backslash\mathcal{GT}$ of geodesics and the critical exponent $δ$ of $Γ$, we show that there exists a constant $C$ depending on $Γ$ and $D$ such that $$\lim_{T\to\infty}μ\left(\left\{[l]\inΓ\backslash\mathcal{GT}\colon \underset{0\le t \le T}{\textrm{max}}d(D,l(t))\le N+y\right\}\right)=e^{-q^y/e^{2δy}}$$ with $$N=\log_{e^{2δ/q}}\left(\frac{T(e^{2δ-q)}}{2e^{2δ}-C(e^{2δ}-q)}\right).$$

math.DS↗

arXiv2018

Notes on the values of the volume entropy

Volume entropy is an important invariant of metric graphs as well as Riemannian manifolds. In this note, we calculate the change of volume entropy when an edge is added to a metric graph. Using the first result, we investigate the change of volume entropy when a vertex and edges around it are added. In the second part, we estimate the value of the volume entropy which can be used to suggest an algorithm of calculating the persistent volume entropy of graphs.

math.DS↗

arXiv2018

Quasi-Sturmian colorings on regular trees

Quasi-Sturmian words, which are infinite words with factor complexity eventually $n+c$ share many properties with Sturmian words. In this paper, we study the quasi-Sturmian colorings on regular trees. There are two different types, bounded and unbounded, of quasi-Sturmian colorings. We obtain an induction algorithm similar to Sturmian colorings. We distinguish them by the recurrence function.

math.DS↗

arXiv2018

Hausdorff dimension in inhomogeneous Diophantine approximation

Let $α$ be an irrational real number. We show that the set of $ε$-badly approximable numbers \[ \mathrm{Bad}^\varepsilon (α) := \{x\in [0,1]\, : \, \liminf_{|q| \to \infty} |q| \cdot \| qα-x \| \geq \varepsilon \} \] has full Hausdorff dimension for some positive $ε$ if and only if $α$ is singular on average. The condition is equivalent to the average $\frac{1}{k} \sum_{i=1, \cdots, k} \log a_i$ of the logarithms of the partial quotients $a_i$ of $α$ going to infinity with $k$. We also consider one-sided approximation, obtain a stronger result when $a_i$ tends to infinity, and establish a partial result in higher dimensions.

math.NT↗

arXiv2018

Volume entropy and information flow in a brain graph

Entropy is a classical measure to quantify the amount of information or complexity of a system. Various entropy-based measures such as functional and spectral entropies have been proposed in brain network analysis. However, they are less widely used than traditional graph theoretic measures such as global and local efficiencies because either they are not well-defined on a graph or difficult to interpret its biological meaning. In this paper, we propose a new entropy-based graph invariant, called volume entropy. It measures the exponential growth rate of the number of paths in a graph, which is a relevant measure if information flows through the graph forever. We model the information propagation on a graph by the generalized Markov system associated to the weighted edge-transition matrix. We estimate the volume entropy using the stationary equation of the generalized Markov system. A prominent advantage of using the stationary equation is that it assigns certain distribution of weights on the edges of the brain graph, which we call the stationary distribution. The stationary distribution shows the information capacity of edges and the direction of information flow on a brain graph. The simulation results show that the volume entropy distinguishes the underlying graph topology and geometry better than the existing graph measures. In brain imaging data application, the volume entropy of brain graphs was significantly related to healthy normal aging from 20s to 60s. In addition, the stationary distribution of information propagation gives a new insight into the information flow of functional brain graph.

q-bio.NC↗

arXiv2017

Asymptotic distribution of values of isotropic quadratic forms at $S$-integral points

We prove an analogue of a theorem of Eskin-Margulis-Mozes: suppose we are given a finite set of places $S$ over $\mathbb{Q}$ containing the archimedean place and excluding the prime $2$, an irrational isotropic form ${\mathbf q}$ of rank $n\geq 4$ on $\mathbb{Q}_S$, a product of $p$-adic intervals $I_p$, and a product $Ω$ of star-shaped sets. We show that unless $n=4$ and ${\mathbf q}$ is split in at least one place, the number of $S$-integral vectors ${\mathbf v} \in {\mathsf{T}} Ω$ satisfying simultaneously ${\mathbf q}( {\mathbf v} ) \in I_p$ for $p \in S$ is asymptotically given by $$ λ({\mathbf q}, Ω) | I| \cdot \prod_{p\in S_f} T_p^{n-2},$$ as ${\mathsf{T}}$ goes to infinity, where $| I |$ is the product of Haar measures of the $p$-adic intervals $I_p$. The proof uses dynamics of unipotent flows on $S$-arithmetic homogeneous spaces; in particular, it relies on an equidistribution result for certain translates of orbits applied to test functions with a controlled growth at infinity, specified by an $S$-arithmetic variant of the $ α$-function introduced in the work of Eskin, Margulis, Mozes, and an $S$-arithemtic version of a theorem of Dani-Margulis.

math.DS↗

arXiv2017

Dimension bound for badly approximable grids

We show that for almost any vector $v$ in $\mathbb{R}^n$, for any $ε>0$ there exists $δ>0$ such that the dimension of the set of vectors $w$ satisfying $\liminf_{k\to\infty} k^{1/n} \ge ε$ (where $<\cdot>$ denotes the distance from the nearest integer), is bounded above by $n-δ$. This result is obtained as a corollary of a discussion in homogeneous dynamics and the main tool in the proof is a relative version of the principle of uniqueness of measures with maximal entropy.

math.DS↗

arXiv2017

Equidistribution with an error rate and Diophantine approximation over function fields

We prove pointwise equidistribution with an error rate of each $H$-orbit in $SL(d,\mathbf{K})/SL(d,\mathbf{Z})$ for a certain proper subgroup $H$ of horospherical group over a function field $\mathbf{K}$, extending a work of Kleinbock-Shi-Weiss. Moreover, we obtain an asymptotic formula for the number of integral solutions to the Diophantine inequalities with weights, generalizing a result of Dodson-Kristensen-Levesley. This result enables us to show pointwise equidistribution for unbounded functions of class $C_α$, which was first introduced by Eskin-Margulis-Mozes.

math.DS↗