arXiv subjects

Derong Kong

Publications and source records attributed to Derong Kong.

At least 37 records · Page 2Linked to original sources

arXiv2023

On the union of homogeneous symmetric Cantor set with its translations

Fix a positive integer $N$ and a real number $0< β< 1/(N+1)$. Let $Γ$ be the homogeneous symmetric Cantor set generated by the IFS $$ \Big\{ ϕ_i(x)=βx + i \frac{1-β}{N}: i=0,1,\cdots, N \Big\}. $$ For $m\in\mathbb{Z}_+$ we show that there exist infinitely many translation vectors $\mathbf t=(t_0,t_1,\cdots, t_m)$ with $0=t_0<t_1<\cdots<t_m$ such that the union $\bigcup_{j=0}^m(Γ+t_j)$ is a self-similar set. Furthermore, for $0< β< 1/(2N+1)$, we give a complete characterization on which the union $\bigcup_{j=0}^m(Γ+t_j)$ is a self-similar set. Our characterization relies on determining whether some related directed graph has no cycles, or whether some related adjacency matrix is nilpotent.

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arXiv2022

Density spectrum of Cantor measure

Given $ρ\in(0, 1/3]$, let $μ$ be the Cantor measure satisfying $μ=\frac{1}{2}μf_0^{-1}+\frac{1}{2}μf_1^{-1}$, where $f_i(x)=ρx+i(1-ρ)$ for $i=0, 1$. The support of $μ$ is a Cantor set $C$ generated by the iterated function system $\{f_0, f_1\}$. Continuing the work of Feng et al. (2000) on the pointwise lower and upper densities \[ Θ_*^s(μ, x)=\liminf_{r\to 0}\frac{μ(B(x,r))}{(2r)^s},\qquad Θ^{*s}(μ, x)=\limsup_{r\to 0}\frac{μ(B(x,r))}{(2r)^s}, \] where $s=-\log 2/\logρ$ is the Hausdorff dimension of $C$, we give a complete description of the sets $D_*$ and $D^*$ consisting of all possible values of the lower and upper densities, respectively. We show that both sets contain infinitely many isolated and infinitely many accumulation points, and they have the same Hausdorff dimension as the Cantor set $C$. Furthermore, we compute the Hausdorff dimension of the level sets of the lower and upper densities. Our method consists in formulating an equivalent ``dyadic" version of the problem involving the doubling map on $[0,1)$, which we solve by using known results on the entropy of a certain open dynamical system and the notion of tuning.

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arXiv2022

Univoque bases of real numbers: simply normal bases, irregular bases and multiple rationals

Given a positive integer $M$ and a real number $x\in(0,1]$, we call $q\in(1,M+1]$ a univoque simply normal base of $x$ if there exists a unique simply normal sequence $(d_i)\in\{0,1,\ldots,M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i q^{-i}$. Similarly, a base $q\in(1,M+1]$ is called a univoque irregular base of $x$ if there exists a unique sequence $(d_i)\in\{0,1,\ldots, M\}^\mathbb N$ such that $x=\sum_{i=1}^\infty d_i q^{-i}$ and the sequence $(d_i)$ has no digit frequency. Let $\mathcal U_{SN}(x)$ and $\mathcal U_{I_r}(x)$ be the sets of univoque simply normal bases and univoque irregular bases of $x$, respectively. In this paper we show that for any $x\in(0,1]$ both $\mathcal U_{SN}(x)$ and $\mathcal U_{I_r}(x)$ have full Hausdorff dimension. Furthermore, given finitely many rationals $x_1, x_2, \ldots, x_n\in(0,1]$ so that each $x_i$ has a finite expansion in base $M+1$, we show that there exists a full Hausdorff dimensional set of $q\in(1,M+1]$ such that each $x_i$ has a unique expansion in base $q$.

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arXiv2022

Intersections of middle-$α$ Cantor sets with a fixed translation

For $λ\in(0,1/3]$ let $C_λ$ be the middle-$(1-2λ)$ Cantor set in $\mathbb R$. Given $t\in[-1,1]$, excluding the trivial case we show that \[ Λ(t):=\left\{λ\in(0,1/3]: C_λ\cap(C_λ+t)\ne\emptyset\right\} \] is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of $Λ(t)$, which reveals a dimensional variation principle. Furthermore, for any $β\in[0,1]$ we show that the level set \[ Λ_β(t):=\left\{λ\inΛ(t): \dim_H(C_λ\cap(C_λ+t))=\dim_P(C_λ\cap(C_λ+t))=β\frac{\log 2}{-\log λ}\right\} \] has equal Hausdorff and packing dimension $(-β\logβ-(1-β)\log\frac{1-β}{2})/\log 3$. We also show that the set of $λ\inΛ(t)$ for which $\dim_H(C_λ\cap(C_λ+t))\ne\dim_P(C_λ\cap(C_λ+t))$ has full Hausdorff dimension.

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arXiv2021

How likely can a point be in different Cantor sets

Let $m\in\mathbb N_{\ge 2}$, and let $\mathcal K=\{K_λ: λ\in(0, 1/m]\}$ be a class of Cantor sets, where $K_λ=\{\sum_{i=1}^\infty d_iλ^i: d_i\in\{0,1,\ldots, m-1\}, i\ge 1\}$. We investigate in this paper the likelyhood of a fixed point in the Cantor sets of $\mathcal K$. More precisely, for a fixed point $x\in(0,1)$ we consider the parameter set $Λ(x)=\{λ\in(0,1/m]: x\in K_λ\}$, and show that $Λ(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets with large thickness in $Λ(x)$ we prove that the intersection $Λ(x)\capΛ(y)$ also has full Hausdorff dimension for any $x, y\in(0,1)$.

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arXiv2020

On the smallest base in which a number has a unique expansion

Given a real number $x>0$, we determine $q_s(x):=\inf\mathscr{U}(x)$, where $\mathscr{U}(x)$ is the set of all bases $q\in(1,2]$ for which $x$ has a unique expansion of $0$'s and $1$'s. We give an explicit description of $q_s(x)$ for several regions of $x$-values. For others, we present an efficient algorithm to determine $q_s(x)$ and the lexicographically smallest unique expansion of $x$. We show that the infimum is attained for almost all $x$, but there is also a set of points of positive Hausdorff dimension for which the infimum is proper. In addition, we show that the function $q_s$ is right-continuous with left-hand limits and no downward jumps, and characterize the points of discontinuity of $q_s$. A large part of the paper is devoted to the level sets $L(q):=\{x>0:q_s(x)=q\}$. We show that $L(q)$ is finite for almost every $q$, but there are also infinitely many infinite level sets. In particular, for the Komornik-Loreti constant $q_{KL}=\min\mathscr{U}(1)\approx 1.787$ we prove that $L(q_{KL})$ has both infinitely many left- and infinitely many right accumulation points.

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arXiv2020

Univoque bases of real numbers: local dimension, Devil's staircase and isolated points

Given a positive integer $M$ and a real number $x>0$, let $\mathcal U(x)$ be the set of all bases $q\in(1, M+1]$ for which there exists a unique sequence $(d_i)=d_1d_2\ldots$ with each digit $d_i\in\{0,1,\ldots, M\}$ satisfying $$ x=\sum_{i=1}^\infty\frac{d_i}{q^i}. $$ The sequence $(d_i)$ is called a $q$-expansion of $x$. In this paper we investigate the local dimension of $\mathcal U(x)$ and prove a `variation principle' for unique non-integer base expansions. We also determine the critical values of $\mathcal U(x)$ such that when $x$ passes the first critical value the set $\mathcal U(x)$ changes from a set with positive Hausdorff dimension to a countable set, and when $x$ passes the second critical value the set $\mathcal U(x)$ changes from an infinite set to a singleton. Denote by $\mathbf U(x)$ the set of all unique $q$-expansions of $x$ for $q\in\mathcal U(x)$. We give the Hausdorff dimension of $\mathbf U(x)$ and show that the dimensional function $x\mapsto\dim_H\mathbf U(x)$ is a non-increasing Devil's staircase. Finally, we investigate the topological structure of $\mathcal U(x)$. In contrast with $x=1$ that $\mathcal U(1)$ has no isolated points, we prove that for typical $x>0$ the set $\mathcal U(x)$ contains isolated points.

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arXiv2020

On a kind of self-similar sets with complete overlaps

Let $E$ be the self-similar set generated by the {\it iterated function system} {\[ f_0(x)=\frac{x}β,\quad f_1(x)=\frac{x+1}β, \quad f_{β+1}=\frac{x+β+1}β \]}with $β\ge 3$. {Then} $E$ is a self-similar set with complete {overlaps}, i.e., $f_{0}\circ f_{β+1}=f_{1}\circ f_1$, but $E$ is not totally self-similar. We investigate all its generating iterated function systems, give the spectrum of $E$, and determine the Hausdorff dimension and Hausdorff measure of $E$ and of the sets which contain all points in $E$ having finite or infinite different triadic codings.

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arXiv2020

Pointwise densities of homogeneous Cantor measure and critical values

Let $N\ge 2$ and $ρ\in(0,1/N^2]$. The homogenous Cantor set $E$ is the self-similar set generated by the iterated function system \[ \left\{f_i(x)=ρx+\frac{i(1-ρ)}{N-1}: i=0,1,\ldots, N-1\right\}. \] Let $s=\dim_H E$ be the Hausdorff dimension of $E$, and let $μ=\mathcal H^s|_E$ be the $s$-dimensional Hausdorff measure restricted to $E$. In this paper we describe, for each $x\in E$, the pointwise lower $s$-density $Θ_*^s(μ,x)$ and upper $s$-density $Θ^{*s}(μ, x)$ of $μ$ at $x$. This extends some early results of Feng et al. (2000). Furthermore, we determine two critical values $a_c$ and $b_c$ for the sets \[ E_*(a)=\left\{x\in E: Θ_*^s(μ, x)\ge a\right\}\quad\textrm{and}\quad E^*(b)=\left\{x\in E: Θ^{*s}(μ, x)\le b\right\} \] respectively, such that $\dim_H E_*(a)>0$ if and only if $a 0$ if and only if $b>b_c$. We emphasize that both values $a_c$ and $b_c$ are related to the Thue-Morse type sequences, and our strategy to find them relies on ideas from open dynamics and techniques from combinatorics on words.

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arXiv2019

Multiple codings for self-similar sets with overlaps

In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $Φ=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ on the real line, for each point $x\in E$ we can find a sequence $(i_k)=i_1i_2\ldots\in\{1,\ldots,m\}^\mathbb N$, called a coding of $x$, such that $$ x=\lim_{n\to\infty}f_{i_1}\circ f_{i_{2}}\circ\cdots\circ f_{i_n}(0). $$ For $k=1,2,\ldots, \aleph_0$ or $2^{\aleph_0}$ we investigate the subset $\mathcal U_k(Φ)$ which consists of all $x\in E$ having precisely $k$ different codings. Among several equivalent characterizations we show that $\mathcal U_1(Φ)$ is closed if and only if $\mathcal U_{\aleph_0}(Φ)$ is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of $\mathcal U_k(Φ)$, and show that the corresponding Hausdorff measure of $\mathcal U_k(Φ)$ is always infinite for any $k\ge 2$. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in $\mathcal U_k(Φ)$ and ${U_{\aleph_0}(Φ)}$.

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arXiv2019

Two bifurcation sets arising from the beta transformation with a hole at $0$

Given $β\in(1,2],$ the $β$-transformation $T_β: x\mapsto βx\pmod 1$ on the circle $[0, 1)$ with a hole $[0, t)$ was investigated by Kalle et al.~(2019). They described the set-valued bifurcation set \[ \mathcal E_β:=\{t\in[0, 1): K_β(t')\ne K_β(t)~\forall t'>t\}, \] where $K_β(t):=\{x\in[0, 1): T_β^n(x)\ge t~\forall n\ge 0\}$ is the survivor set. In this paper we investigate the dimension bifurcation set \[ \mathcal B_β:=\{t\in[0, 1): \dim_H K_β(t')\ne \dim_H K_β(t)~\forall t'>t\}, \] where $\dim_H$ denotes the Hausdorff dimension. We show that if $β\in(1,2]$ is a multinacci number then the two bifurcation sets $\mathcal B_β$ and $\mathcal E_β$ coincide. Moreover we give a complete characterization of these two sets. As a corollary of our main result we prove that for $β$ a multinacci number we have $\dim_H(\mathcal E_β\cap[t, 1])=\dim_H K_β(t)$ for any $t\in[0, 1)$. This confirms a conjecture of Kalle et al.~for $β$ a multinacci number.

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arXiv2018

Critical base for the unique codings of fat Sierpinski gasket

Given $β\in(1,2)$ the fat Sierpinski gasket $\mathcal S_β$ is the self-similar set in $\mathbb R^2$ generated by the iterated function system (IFS) \[ f_{β,d}(x)=\frac{x+d}β,\quad d\in\mathcal A:=\{(0, 0), (1,0), (0,1)\}. \] Then for each point $P\in\mathcal S_β$ there exists a sequence $(d_i)\in\mathcal A^\mathbb N$ such that $P=\sum_{i=1}^\infty d_i/β^i$, and the infinite sequence $(d_i)$ is called a \emph{coding} of $P$. In general, a point in $\mathcal S_β$ may have multiple codings since the overlap region $\mathcal O_β:=\bigcup_{c,d\in\mathcal A, c\ne d}f_{β,c}(Δ_β)\cap f_{β,d}(Δ_β)$ has non-empty interior, where $Δ_β$ is the convex hull of $\mathcal S_β$. In this paper we are interested in the invariant set \[ \widetilde{\mathcal U}_β:=\left\{\sum_{i=1}^\infty \frac{d_i}{β^i}\in \mathcal S_β: \sum_{i=1}^\infty\frac{d_{n+i}}{β^i}\notin\mathcal O_β~\forall n\ge 0\right\}. \] Then each point in $ \widetilde{\mathcal U}_β$ has a unique coding. We show that there is a transcendental number $β_c\approx 1.55263$ related to the Thue-Morse sequence, such that $\widetilde{\mathcal U}_β$ has positive Hausdorff dimension if and only if $β>β_{c}$. Furthermore, for $β=β_c$ the set $\widetilde{\mathcal U}_β$ is uncountable but has zero Hausdorff dimension, and for $β<β_c$ the set $\widetilde{\mathcal U}_β$ is at most countable. Consequently, we also answer a conjecture of Sidorov (2007). Our strategy is using combinatorics on words based on the lexicographical characterization of $\widetilde{\mathcal U}_β$.

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arXiv2018

On the structure of $λ$-Cantor set with overlaps

Given $λ\in(0, 1)$, let $E_λ$ be the self-similar set generated by the iterated function system $\{x/3,(x+λ)/3,(x+2)/3\}$. Then $E_λ$ is a self-similar set with overlaps. We obtain the necessary and sufficient condition for $E_λ$ to be totally self-similar, which is a concept first introduced by Broomhead, Montaldi, and Sidorov in 2004. When $E_λ$ is totally self-similar, all its generating IFSs are investigated, and the size of the set of points having finite triadic codings is determined. Besides, we give some properties of the spectrum of $E_λ$ and show that the spectrum of $E_λ$ vanishes if and only if $λ$ is irrational.

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arXiv2018

On the complexity of the set of codings for self-similar sets and a variation on the construction of Champernowne

Let $F=\{\mathbf{p}_0,\ldots,\mathbf{p}_n\}$ be a collection of points in $\mathbb{R}^d.$ The set $F$ naturally gives rise to a family of iterated function systems consisting of contractions of the form $$S_i(\mathbf{x})=λ\mathbf{x} +(1-λ)\mathbf{p}_i,$$ where $λ\in(0,1)$. Given $F$ and $λ$ it is well known that there exists a unique non-empty compact set $X$ satisfying $X=\cup_{i=0}^n S_i(X)$. For each $\mathbf{x} \in X$ there exists a sequence $\mathbf{a}\in\{0,\ldots,n\}^{\mathbb{N}}$ satisfying $$\mathbf{x}=\lim_{j\to\infty}(S_{a_1}\circ \cdots \circ S_{a_j})(\mathbf{0}).$$ We call such a sequence a coding of $\mathbf{x}$. In this paper we prove that for any $F$ and $k \in\mathbb{N},$ there exists $δ_k(F)>0$ such that if $λ\in(1-δ_k(F),1),$ then every point in the interior of $X$ has a coding which is $k$-simply normal. Similarly, we prove that there exists $δ_{uni}(F)>0$ such that if $λ\in(1-δ_{uni}(F),1),$ then every point in the interior of $X$ has a coding containing all finite words. For some specific choices of $F$ we obtain lower bounds for $δ_k(F)$ and $δ_{uni}(F)$. We also prove some weaker statements that hold in the more general setting when the similarities in our iterated function systems exhibit different rates of contraction. Our proofs rely on a variation of a well known construction of a normal number due to Champernowne, and an approach introduced by Erdős and Komornik.

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arXiv2018

Relative bifurcation sets and the local dimension of univoque bases

Fix an alphabet $A=\{0,1,\dots,M\}$ with $M\in\mathbb{N}$. The univoque set $\mathscr{U}$ of bases $q\in(1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper investigates how the set $\mathscr{U}$ is distributed over the interval $(1,M+1)$ by determining the limit $$f(q):=\lim_{δ\to 0}\dim_H\big(\mathscr{U}\cap(q-δ,q+δ)\big)$$ for all $q\in(1,M+1)$. We show in particular that $f(q)>0$ if and only if $q\in\overline{\mathscr{U}}\backslash\mathscr{C}$, where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called {\emph relative bifurcation sets}, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al.~[{\emph arXiv:1612.07982; to appear in Acta Arithmetica}, 2018]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [{\emph Adv. Math.}, 308:575--598, 2017] about strongly univoque sets.

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arXiv2018

Multiple expansions of real numbers with digits set $\{0,1,q\}$

For $q>1$ we consider expansions in base $q$ over the alphabet $\{0,1,q\}$. Let $\mathcal{U}_q$ be the set of $x$ which have a unique $q$-expansions. For $k=2, 3,\cdots,\aleph_0$ let $\mathcal{B}_k$ be the set of bases $q$ for which there exists $x$ having $k$ different $q$-expansions, and for $q\in \mathcal{B}_k$ let $\mathcal{U}_q^{(k)}$ be the set of all such $x$'s which have $k$ different $q$-expansions. In this paper we show that \[ \mathcal{B}_{\aleph_0}=[2,\infty),\quad \mathcal{B}_k=(q_c,\infty)\quad \textrm{for any}\quad k\ge 2, \] where $q_c\approx 2.32472$ is the appropriate root of $x^3-3x^2+2x-1=0$. Moreover, we show that for any positive integer $k\ge 2$ and any $q\in\mathcal{B}_{k}$ the Hausdorff dimensions of $\mathcal{U}_q^{(k)}$ and $\mathcal{U}_q$ are the same, i.e., \[ \dim_H\mathcal{U}_q^{(k)}=\dim_H\mathcal{U}_q\quad\textrm{for any}\quad k\ge 2. \] Finally, we conclude that the set of $x$ having a continuum of $q$-expansions has full Hausdorff dimension.

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arXiv2018

On the bifurcation set of unique expansions

Given a positive integer $M$, for $q\in(1, M+1]$ let ${\mathcal{U}}_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion with the digit set $\{0, 1,\ldots, M\}$, and let $\mathbf{U}_q$ be the set of corresponding $q$-expansions. Recently, Komornik et al.~(Adv. Math., 2017) showed that the topological entropy function $H: q \mapsto h_{top}(\mathbf{U}_q)$ is a Devil's staircase in $(1, M+1]$. Let $\mathcal{B}$ be the bifurcation set of $H$ defined by \[ \mathcal{B}=\{q\in(1, M+1]: H(p)\ne H(q)\quad\textrm{for any}\quad p\ne q\}. \] In this paper we analyze the fractal properties of $\mathcal{B}$, and show that for any $q\in \mathcal{B}$, \[ \lim_{δ\rightarrow 0} \dim_H(\mathcal{B}\cap(q-δ, q+δ))=\dim_H\mathcal{U}_q, \] where $\dim_H$ denotes the Hausdorff dimension. Moreover, when $q\in\mathcal{B}$ the univoque set $\mathcal{U}_q$ is dimensionally homogeneous, i.e., $ \dim_H(\mathcal{U}_q\cap V)=\dim_H\mathcal{U}_q $ for any open set $V$ that intersect $\mathcal{U}_q$. As an application we obtain a dimensional spectrum result for the set $\mathcal{U}$ containing all bases $q\in(1, M+1]$ such that $1$ admits a unique $q$-expansion. In particular, we prove that for any $t>1$ we have \[ \dim_H(\mathcal{U}\cap(1, t])=\max_{ q\le t}\dim_H\mathcal{U}_q. \] We also consider the variations of the sets $\mathcal{U}=\mathcal{U}(M)$ when $M$ changes.

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arXiv2018

Bases in which some numbers have exactly two expansions

In this paper we answer several questions raised by Sidorov on the set $\mathcal B_2$ of bases in which there exist numbers with exactly two expansions. In particular, we prove that the set $\mathcal B_2$ is closed, and it contains both infinitely many isolated and accumulation points in $(1, q_{KL})$, where $q_{KL}\approx 1.78723$ is the Komornik-Loreti constant. Consequently we show that the second smallest element of $\mathcal B_2$ is the smallest accumulation point of $\mathcal B_2$. We also investigate the higher order derived sets of $\mathcal B_2$. Finally, we prove that there exists a $δ>0$ such that \begin{equation*} \dim_H(\mathcal B_2\cap(q_{KL}, q_{KL}+δ))<1, \end{equation*} where $\dim_H$ denotes the Hausdorff dimension.

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