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Dong Han Kim

Publications and source records attributed to Dong Han Kim.

At least 19 recordsLinked to original sources
arXiv2025

On the irrationality exponent of real numbers with low complexity expansion

Let $\xi$ be a real number and $b \ge 2$ an integer. We study the relationship between the irrationality exponent of $\xi$ and the subword complexity $p(n, \mathbf{x})$ of the $b$-ary expansion $\mathbf{x}$ of $\xi$, where $p(n, \mathbf{x})$ counts the number of distinct blocks of length $n$ in $\mathbf{x}$, for $n \ge 1$. If the irrationality exponent of $\xi$ is equal to $2$, which is the case for almost all real numbers $\xi$, we show that the limit superior of the sequence $(p(n, \mathbf{x}) / n)_{n \ge 1}$ is at least equal to 4/3. The proof is based on a careful study of the evolution of the Rauzy graphs of infinite words of low complexity.

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arXiv2025

On the Markoff spectrum on the Hecke group of index six

The discrete part of the Markoff spectrum on the Hecke group of index 6 was determined by A.~Schmidt. In this paper, we study its Markoff and Lagrange spectra after the smallest accumulation point $4/\sqrt3$. We show that both the Markoff and Lagrange spectra below $4/\sqrt{3} + \epsilon$ have positive Hausdorff dimension for any positive $\epsilon$. We also find maximal gaps and an isolated point in the spectra.

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arXiv2025

Exponential Sums by Irrationality Exponent

In this article, we give an asymptotic bound for the exponential sum of the Möbius function $\sum_{n \le x} μ(n) e(αn)$ for a fixed irrational number $α\in\mathbb{R}$. This exponential sum was originally studied by Davenport and he obtained an asymptotic bound of $x(\log x)^{-A}$ for any $A\ge0$. Our bound depends on the irrationality exponent $η$ of $α$. If $η\le 5/2$, we obtain a bound of $x^{4/5 + \varepsilon}$ and, when $η\ge 5/2$, our bound is $x^{(2η-1)/2η+ \varepsilon}$. This result extends a result of Murty and Sankaranarayanan, who obtained the same bound in the case $η= 2$.

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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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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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arXiv2023

Intrinsic Diophantine approximation on circles and spheres

We study Lagrange spectra arising from intrinsic Diophantine approximation of circles and spheres. More precisely, we consider three circles embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$ and three spheres embedded in $\mathbb{R}^3$ or $\mathbb{R}^4$. We present a unified framework to connect the Lagrange spectra of these six spaces with the spectra of $\mathbb{R}$ and $\mathbb{C}$. Thanks to prior work of Asmus L.~Schmidt on the spectra of $\mathbb{R}$ and $\mathbb{C}$, 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.

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arXiv2022

The Markoff and Lagrange spectra on the Hecke group H4

We consider the Markoff spectrum and the Lagrange spectrum on the Hecke group $\mathbf H_4$. They are identical to the Markoff and Lagrange spectra of the unit circle. The Markoff spectrum on $\mathbf H_4$ is also known as the Markoff spectrum of index 2 sublattices by Vulakh and the Markoff spectrum of 2-minimal forms or $C$-minimal forms by Schmidt. They characterized the spectrum up to the first accumulation point, independently. We show that, after the first accumulation point, both spectra have positive Hausdorff dimension. Then we find gaps in the spectra and give a bound on Hall's ray.

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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

On the multiple recurrence properties for disjoint systems

We consider mutually disjoint family of measure preserving transformations $T_1, \cdots, T_k$ on a probability space $(X, \mathcal{B}, μ)$. We obtain the multiple recurrence property of $T_1, \cdots, T_k$ and this result is utilized to derive multiple recurrence of Poincaré type in metric spaces. We also present multiple recurrence property of Khintchine type. Further, we study multiple ergodic averages of disjoint systems and we show that $T_1, \cdots, T_k$ are uniformly jointly ergodic if each $T_i$ is ergodic.

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arXiv2021

Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum

Let $\mathscr{L}(S^1)$ 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 $\mathscr{L}(S^1)$ 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 $\mathscr{L}(S^1)$ 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.

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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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arXiv2019

Generic point equivalence and Pisot numbers

Let $β>1$ be an integer or generally a Pisot number. Put $T(x) = \{ βx \}$ on $[0,1]$ and let $S: [0,1]\to [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm β^m$ with positive integers $m$. We give sufficient conditions that $T$ and $S$ have the same generic points.

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arXiv2019

Number theoretical properties of Romik's dynamical system

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 field of rationals.

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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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arXiv2018

Some constructions for the higher-dimensional three-distance theorem

For a given real number $α$, let us place the fractional parts of the points $0, α, 2 α,$ $ \cdots, (N-1) α$ on the unit circle. These points partition the unit circle into intervals having at most three lengths, one being the sum of the other two. This is the three distance theorem. We consider a two-dimensional version of the three distance theorem obtained by placing on the unit circle the points $ nα+ mβ$, for $0 \leq n,m < N$. We provide examples of pairs of real numbers $(α,β)$, with $1,α, β$ rationally independent, for which there are finitely many lengths between successive points (and in fact, seven lengths), with $(α,β)$ not badly approximable, as well as examples for which there are infinitely many lengths.

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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.

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