arXiv subjects

Seul Bee Lee

Publications and source records attributed to Seul Bee Lee.

12 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 $Θ\backslash\mathbb{H}^2$ associated with the theta group $Θ$. 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 $Θ\backslash\mathbb{H}^2$.

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arXiv2025

Moment formulas of Siegel transforms with congruence conditions in dimension 2

We compute the first and second moment formulas for Siegel transforms related to problems counting primitive lattice points in the real plane with congruence conditions. As applications, we derive an analog of Schmidt's random counting theorem and the quantitative Khintchine theorem for irrational numbers, approximated by rational numbers $p/q$, where we place a congruence-conditional constraint on the vector $(p,q)$.

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arXiv2025

Uniform Diophantine approximation on the Hecke group $\mathbf H_4$

Dirichlet's uniform approximation theorem is a fundamental result in Diophantine approximation that gives an optimal rate of approximation with a given bound. We study uniform Diophantine approximation properties on the Hecke group $\mathbf H_4$. For a given real number $α$, we characterize the sequence of $\mathbf H_4$-best approximations of $α$ and show that they are convergents of the Rosen continued fraction and the dual Rosen continued fraction of $α$. We give analogous theorems of Dirichlet uniform approximation and the Legendre theorem with optimal constants.

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arXiv2025

The Brjuno and Wilton Functions

The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function $B(x)$ is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function $W(x)$ stems from the study of divisor sums and self-correlation functions in analytic number theory. We show that these perspectives are unified by the semi-Brjuno function $B_0(x)$. Namely, $B(x)$ and $W(x)$ can be expressed in terms of the even and odd parts of $B_0(x)$, respectively, up to a bounded defect. Based on numerical observations, we further analyze the arising functions $Δ^+(x) = B^+(x) - 2B_0^+(x)$ and $Δ^-(x) = W^-(x) - 2B_0^-(x)$, the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.

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arXiv2024

Regularity properties of the $α$-Wilton functions

The aim of this article is to study the regularity properties of the Wilton functions $W_α$ associated with $α$-continued fractions. We prove that the Wilton function is BMO for $α\in[1-g,g]$ (where $g:=\frac{\sqrt{5}-1}{2}$ denotes the golden number), and we show that this result is optimal, since we find that on any left neighbourhood of $1-g$ and on any right neighbourhood of $g$ there are values $α$ for which $W_α$ is not BMO; the proof of this latter negative results exploits a special feature of the family of $α$-continued fractions called ``matching''. Our results complete those of Marmi--Moussa--Yoccoz (1997) and of Lee--Marmi--Petrykiewicz--Schindler (2024), where it is proven that Wilton function is BMO for, respectively, $α=1/2$ (\cite{MaMoYo_97}) and $α\in[\frac{1}{2},g]$ (\cite{LeMar_24}).

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arXiv2024

Diophantine approximation by rational numbers of certain parity types

For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.

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arXiv2022

The Brjuno functions of the by-excess, odd, even and odd-odd continued fractions and their regularity properties

The Brjuno function was introduced by Yoccoz to study the linearizability of holomorphic germs and other one-dimensional small divisor problems. The Brjuno functions associated with various continued fractions including the by-excess continued fraction were subsequently investigated: it was conjectured that the difference between the classical Brjuno function and the even part of the Brjuno function associated with the by-excess continued fraction extends to a Hölder continuous function of the whole real line. In this paper, we prove this conjecture and we extend its validity to the more general case of Brjuno functions with positive exponents. Moreover, we study the Brjuno functions associated to the odd and even continued fractions introduced by Schweiger. We show that they belong to all $L^p$ spaces, $p\ge1$. We prove that the Brjuno function associated to the odd continued fraction differs from the classical Brjuno function by a Hölder continuous function. On the other hand, the Brjuno function associated to the even continued fraction differs from the classical Brjuno function by a sum of a Hölder continuous function and a Brjuno-type function associated to the odd-odd continued fraction, introduced in the study of the best approximations of the form odd/odd.

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arXiv2021

A convergence criterion for the unstable manifolds of the MacKay approximate renormalisation

We give an explicit arithmetical condition which guarantees the existence of the unstable manifold of the MacKay approximate renormalisation scheme for the breakup of invariant tori in one and a half degrees of freedom Hamiltonian systems, correcting earlier results. Furthermore, when our condition is violated, we give an example of points on which the unstable manifold does not converge.

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arXiv2021

On the Lévy constants of Sturmian continued fractions

The Lévy constant of an irrational real number is defined by the exponential growth rate of the sequence of denominators of the principal convergents in its continued fraction expansion. Any quadratic irrational has an ultimately periodic continued fraction expansion and it is well-known that this implies the existence of a Lévy constant. Let $a, b$ be distinct positive integers. If the sequence of partial quotients of an irrational real number is a Sturmian sequence over $\{a, b\}$, then it has a Lévy constant, which depends only on $a$, $b$, and the slope of the Sturmian sequence, but not on its intercept. We show that the set of Lévy constants of irrational real numbers whose sequence of partial quotients is periodic or Sturmian is equal to the whole interval $[\log ((1+\sqrt 5)/2 ), + \infty)$.

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arXiv2021

Regularity properties of $k$-Brjuno and Wilton functions

We study functions related to the classical Brjuno function, namely $k$-Brjuno functions and the Wilton function. Both appear in the study of boundary regularity properties of (quasi) modular forms and their integrals. We consider various possible versions of them, based on the $α$-continued fraction developments. We study their BMO regularity properties and their behaviour near rational numbers of their finite truncations.

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arXiv2020

Odd-odd continued fraction algorithm

By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among others, it is proved that all the best approximating rationals of odd denominators and odd numerators of an irrational number are given by the principal convergents of the odd-odd continued fraction algorithm and vice versa.

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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.

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