arXiv subjects

Dong Han Kim

Publications and source records attributed to Dong Han Kim.

At least 37 records · Page 2Linked to original sources

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

Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential

Let $((0,1], T)$ be the doubling map in the unit interval and $φ$ be the Saint-Petersburg potential, defined by $φ(x)=2^n$ if $x\in (2^{-n-1}, 2^{-n}]$ for all $n\geq 0$. We consider the asymptotic properties of the Birkhoff sum $S\_n(x)=φ(x)+\cdots+φ(T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{n\log n}S\_n(x)$ converges to $\frac{1}{\log 2}$ in probability. We determine the Hausdorff dimension of the level set $\{x: \lim\_{n\to\infty}S\_n(x)/n=α\} \ (α>0)$, as well as that of the set $\{x: \lim\_{n\to\infty}S\_n(x)/Ψ(n)=α\} \ (α>0)$, when $Ψ(n)=n\log n, n^a $ or $2^{n^γ}$ for $a>1$, $γ>0$. The fast increasing Birkhoff sum of the potential function $x\mapsto 1/x$ is also studied.

math.DS↗

arXiv2017

A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let $b \ge 2$ be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of $b$-ary digits is a Sturmian sequence over $\{0,1,\ldots, b-1\}$ and we prove that this lower bound is best possible. As an application, we derive some information on the $b$-ary expansion of $\log(1+\frac{1}{a})$,for any integer $a \ge 34$.

math.NT↗

arXiv2017

Hausdorff dimension of the set approximated by irrational rotations

Let $θ$ be an irrational number and $φ: {\mathbb N} \to {\mathbb R}^{+}$ be a monotone decreasing function tending to zero. Let $$E_φ(θ) =\Big\{y \in \mathbb R: \|nθ- y\|<φ(n), \ {\text{for infinitely many}}\ n\in {\mathbb N} \Big\}, $$ i.e. the set of points which are approximated by the irrational rotation with respect to the error function $φ(n)$. In this article, we give a complete description of the Hausdorff dimension of $E_φ(θ)$ for any monotone function $φ$ and any irrational $θ$.

math.NT↗

arXiv2017

Dirichlet uniformly well-approximated numbers

Fix an irrational number $θ$. For a real number $τ>0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\leq n\leq Q$, such that $\|nθ-y\| 0$, the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of $θ$. It is also proved that with respect to $τ$, the only possible discontinuous point of the Hausdorff dimension is $τ=1$.

math.NT↗

arXiv2017

Long Hitting time for translation flows and L-shaped billiards

We consider the flow in direction $θ$ on a translation surface and we study the asymptotic behavior for $r\to 0$ of the time needed by orbits to hit the $r$-neighborhood of a prescribed point, or more precisely the exponent of the corresponding power law, which is known as hitting time. For flat tori the limsup of hitting time is equal to the diophantine type of the direction $θ$. In higher genus, we consider a generalized geometric notion of diophantine type of a direction $θ$ and we seek for relations with hitting time. For genus two surfaces with just one conical singularity we prove that the limsup of hitting time is always less or equal to the square of the diophantine type. For any square-tiled surface with the same topology the diophantine type itself is a lower bound, and any value between the two bounds can be realized, moreover this holds also for a larger class of origamis satisfying a specific topological assumption. Finally, for the so-called Eierlegende Wollmilchsau origami, the equality between limsup of hitting time and diophantine type subsists. Our results apply to L-shaped billiards.

math.DS↗

arXiv2017

Continued fraction algorithm for Sturmian colorings of trees

Factor complexity $b_ϕ(n)$ for a vertex coloring $ϕ$ of a regular tree is the number of colored $n$-balls up to color-preserving automorphisms. Sturmian colorings are colorings of minimal unbounded factor complexity $b_ϕ(n) = n+2$. In this article, we prove an induction algorithm for Sturmian colorings using colored balls in a way analogous to induction algorithm of Sturmian words. Furthermore, we characterize Sturmian colorings in terms of the data for the induction algorithm.

math.DS↗

arXiv2016

On the expansions of real numbers in two integer bases

Let $r$ and $s$ be multiplicatively independent positive integers. We establish that the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

math.NT↗

arXiv2016

On the expansions of real numbers in two multiplicative dependent bases

Let $r \ge 2$ and $s \ge 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and of the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0, 1, \ldots , r-1\}$ and $\{0, 1, \ldots , s-1\}$, and we show that this bound is best possible.

math.NT↗

arXiv2016

Zero-one law of Hausdorff dimensions of the recurrent sets

Let $(Σ, σ)$ be the one-sided shift space with $m$ symbols and $R_n(x)$ be the first return time of $x\inΣ$ to the $n$-th cylinder containing $x$. Denote $$E^φ_{α,β}=\left\{x\inΣ: \liminf_{n\to\infty}\frac{\log R_n(x)}{φ(n)}=α,\ \limsup_{n\to\infty}\frac{\log R_n(x)}{φ(n)}=β\right\},$$ where $φ: \mathbb{N}\to \mathbb{R}^+$ is a monotonically increasing function and $0\leqα\leqβ\leq +\infty$. We show that the Hausdorff dimension of the set $E^φ_{α,β}$ admits a dichotomy: it is either zero or one depending on $φ, α$ and $β$.

math.DS↗

arXiv2015

On the $b$-ary expansions of $\log (1 + \frac{1}{a})$ and ${\mathrm e}$

Let $b \ge 2$ be an integer and $ξ$ an irrational real number. We prove that, if the irrationality exponent of $ξ$ is equal to $2$ or slightly greater than $2$, then the $b$-ary expansion of $ξ$ cannot be `too simple', in a suitable sense. Our result applies, among other classical numbers, to badly approximable numbers, non-zero rational powers of ${\mathrm e}$, and $\log (1 + \frac{1}{a})$, provided that the integer $a$ is sufficiently large. It establishes an unexpected connection between the irrationality exponent of a real number and its $b$-ary expansion.

math.NT↗

arXiv2015

On Kurzweil's 0-1 Law in Inhomogeneous Diophantine Approximation

We give a sufficient and necessary condition such that for almost all $s\in{\mathbb R}$ \[ \|nθ-s\|<ψ(n)\qquad\text{for infinitely many}\ n\in{\mathbb N}, \] where $θ$ is fixed and $ψ(n)$ is a positive, non-increasing sequence. This improves upon an old result of Kurzweil and contains several previous results as special cases: two theorems of Kurzweil, a theorem of Tseng and a recent result of the second author. Moreover, we also discuss an analogue of our result in the field of formal Laurent series which has similar consequences.

math.NT↗

arXiv2014

A characterization of eventually periodicity

In this article, we show that the Kamae-Xue complexity function for an infinite sequence classifies eventual periodicity completely. We prove that an infinite binary word $x_1x_2 \cdots $ is eventually periodic if and only if $Σ(x_1x_2\cdots x_n)/n^3$ has a positive limit, where $Σ(x_1x_2\cdots x_n)$ is the sum of the squares of all the numbers of appearance of finite words in $x_1 x_2 \cdots x_n$, which was introduced by Kamae-Xue as a criterion of randomness in the sense that $x_1x_2\cdots x_n$ is more random if $Σ(x_1x_2\cdots x_n)$ is smaller. In fact, it is known that the lower limit of $Σ(x_1x_2\cdots x_n) /n^2 $ is at least 3/2 for any sequence $x_1x_2 \cdots$, while the limit exists as 3/2 almost surely for the $(1/2,1/2)$ product measure. For the other extreme, the upper limit of $Σ(x_1x_2\cdots x_n)/n^3$ is bounded by 1/3. There are sequences which are not eventually periodic but the lower limit of $Σ(x_1x_2\cdots x_n)/n^3$ is positive, while the limit does not exist.

cs.CC↗

arXiv2014

Distribution of the asset price movement and market potential

In this article we discuss the distribution of asset price movements by the market potential function. From the principle of free energy minimization we analyze two different kinds of market potentials. We obtain a U-shaped potential when market reversion (i.e. contrarian investors) is dominant. On the other hand, if there are more trend followers, flat and logarithmic--like potentials appeared. By using the Cyclical Adjusted Price-to-Earning ratio, which is a common valuation tool, we empirically investigate the market data. By studying long term data we observe the historical change of the market potential of the US stock market. Recent US data shows that the market potential looks more likely as the trending followers' potential. Next, we compare the market potentials for 12 different countries. Though some countries have similar market potentials, there are specific examples like Japan which exhibits very flat potential.

q-fin.ST↗

arXiv2014

Farey map, Diophantine approximation and Bruhat-Tits tree

Based on Broise-Alamichel and Paulin's work on the Gauss map corresponding to the principal convergents, we continue the study of the Gauss map via Farey maps to contain all the intermediate convergents. We define the geometric Farey map, which is given by time-1 map of the geodesic flow. We also define algebraic Farey maps, better suited for arithmetic properties, which produce all the intermediate convergents. Then we obtain the ergodic invariant measures for the Farey maps and the convergent speed.

math.DS↗

arXiv2013

On the refined shrinking target property of rotations

We discuss the shrinking target property of irrational rotations. We obtain the condition of an irrational $θ$ and monotone increasing $φ(n)$ such that $$ \liminf_{n \to \infty} n φ(n) \| nθ- s \| = 0 \text{ for almost every } s. $$ We also consider the class of irrationals for which the limit inferior is 0 for every monotone increasing $φ(n)$ such that $\sum_n 1/(nφ(n))$ diverges.

math.NT↗

arXiv2013

Bounded type interval exchange maps

Irrational numbers of bounded type have several equivalent characterizations. They have bounded partial quotients in terms of arithmetic characterization and in the dynamics of the circle rotation, the rescaled recurrence time to $r$-ball of the initial point is bounded below. In this paper, we consider how the bounded type condition of irrational is generalized into interval exchange maps.

math.DS↗

arXiv2012

Subword complexity and Sturmian colorings of regular trees

In this article, we study subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and study Sturmian colorings, which are colorings of minimal unbounded subword complexity. We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray with loops possibly attached, thus a lifting of an "infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones.

math.DS↗