arXiv subjects

Kunkun Song

Publications and source records attributed to Kunkun Song.

7 recordsLinked to original sources

arXiv2025

Multifractal analysis of maximal product of consecutive partial quotients in continued fractions

Let $[a_1(x), a_2(x), \ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0,1)$. We study the growth rate of the maximal product of consecutive partial quotients among the first $n$ terms, defined by $L_n(x)=\max_{1\leq i\leq n}\{a_i(x)a_{i+1}(x)\}$, from the viewpoint of multifractal analysis. More precisely, we determine the Hausdorff dimension of the level set \[L(φ):=\left\{x\in (0,1):\lim_{n\to \infty}\frac{L_n(x)}{φ(n)}=1\right\},\] where $φ:\mathbb{R^+}\to\mathbb{R^+}$ is an increasing function such that $\log φ$ is a regularly increasing function with index $ρ$. We show that there exists a jump of the Hausdorff dimension of $L(φ)$ when $ρ=1/2$. We also construct uncountably many discontinuous functions $ψ$ that cause the Hausdorff dimension of $L(ψ)$ to transition continuously from 1 to 1/2, filling the gap when $ρ=1/2$.

math.NT↗

arXiv2025

Multifractal analysis of the convergence exponents for the digits in $d$-decaying Gauss like dynamical systems

Let $\{a_n(x)\}_{n\geq1}$ be the sequence of digits of $x\in(0,1)$ in infinite iterated function systems with polynomial decay of the derivative. We first study the multifractal spectrum of the convergence exponent defined by the sequence of the digits $\{a_n(x)\}_{n\geq1}$ and the weighted products of distinct digits with finite numbers respectively, and then calculate the Hausdorff dimensions of the intersection of sets defined by the convergence exponent of the weighted product of distinct digits with finite numbers and sets of points whose digits are non-decreasing in such iterated function systems.

math.DS↗

arXiv2024

Multifractal analysis of the growth rate of digits in Schneider's $p$-adic continued fraction dynamical system

Let $\mathbb{Z}_p$ be the ring of $p$-adic integers and $a_n(x)$ be the $n$-th digit of Schneider's $p$-adic continued fraction of $x\in p\mathbb{Z}_p$. We study the growth rate of the digits $\{a_n(x)\}_{n\geq1}$ from the viewpoint of multifractal analysis. The Hausdorff dimension of the set \[E_{\sup}(ψ)=\Big\{x\in p\mathbb{Z}_p:\ \limsup\limits_{n\to\infty}\frac{a_n(x)}{ψ(n)}=1\Big\}\] is completely determined for any $ψ:\mathbb{N}\to\mathbb{R}^{+}$ satisfying $ψ(n)\to \infty$ as $n\to\infty$. As an application, we also calculate the Hausdorff dimension of the intersection sets \[E^{\sup}_{\inf}(ψ,α_1,α_2)=\left\{x\in p\mathbb{Z}_p:\liminf_{n\rightarrow\infty}\dfrac{a_n(x)}{ψ(n)}=α_1,~\limsup_{n\rightarrow\infty}\dfrac{a_n(x)}{ψ(n)}=α_2\right\}\] for the above function $ψ$ and $0\leqα_1<α_2\leq\infty$.

math.NT↗

arXiv2022

Dimensions of certain sets of continued fractions with non-decreasing partial quotients

Let $[a_1(x),a_2(x),a_3(x),\cdots]$ be the continued fraction expansion of $x\in (0,1)$. This paper is concerned with certain sets of continued fractions with non-decreasing partial quotients. As a main result, we obtain the Hausdorff dimension of the set \[\left\{x\in(0,1): a_1(x)\leq a_2(x)\leq \cdots,\ \limsup\limits_{n\to\infty}\frac{\log a_n(x)}{ψ(n)}=1\right\}\] for any $ψ:\mathbb{N}\rightarrow\mathbb{R}^+$ satisfying $ψ(n)\to\infty$ as $n\to\infty$.

math.NT↗

arXiv2020

Some exceptional sets of Borel-Bernstein Theorem in continued fractions

Let $[a_1(x),a_2(x), a_3(x),\cdots]$ denote the continued fraction expansion of a real number $x \in [0,1)$. This paper is concerned with certain exceptional sets of the Borel-Bernstein Theorem on the growth rate of $\{a_n(x)\}_{n\geq1}$. As a main result, the Hausdorff dimension of the set \[ E_{\sup}(ψ)=\left\{x\in[0,1):\ \limsup\limits_{n\to\infty}\frac{\log a_n(x)}{ψ(n)}=1\right\} \] is determined, where $ψ:\mathbb{N}\rightarrow\mathbb{R}^+$ tends to infinity as $n\to\infty$.

math.NT↗

arXiv2018

Fourier decay rate of coin-tossing type measures

The Fourier decay rate of a coin-tossing type measure is investigated. Explicit estimation is given. Our method relies upon a classical result of Hartman and Kershner. As an application, we present a new example of a measure whose Fourier decay rate could be as slowly as we please, and for this measure almost every point is absolutely normal.

math.FA↗

arXiv2016

A remark on the extreme value theory for continued fractions

Let $x$ be a irrational number in the unit interval and denote by its continued fraction expansion $[a_1(x), a_2(x), \cdots, a_n(x), \cdots]$. For any $n \geq 1$, write $T_n(x) = \max_{1 \leq k \leq n}\{a_k(x)\}$. We are interested in the Hausdorff dimension of the fractal set \[ E_ϕ= \left\{x \in (0,1): \lim_{n \to \infty} \frac{T_n(x)}{ϕ(n)} =1\right\}, \] where $ϕ$ is a positive function defined on $\mathbb{N}$ with $ϕ(n) \to \infty$ as $n \to \infty$. Some partial results have been obtained by Wu and Xu, Liao and Rams, and Ma. In the present paper, we further study this topic when $ϕ(n)$ tends to infinity with a doubly exponential rate as $n$ goes to infinity.

math.NT↗