arXiv subjects

Derong Kong

Publications and source records attributed to Derong Kong.

At least 37 records · Page 2Linked to original sources

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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arXiv2018

Bifurcation sets arising from non-integer base expansions

Given a positive integer $M$ and $q\in(1,M+1]$, let $\mathcal U_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion: there exists a unique sequence $(x_i)=x_1x_2\ldots$ with each $x_i\in\{0,1,\ldots, M\}$ such that \[ x=\frac{x_1}{q}+\frac{x_2}{q^2}+\frac{x_3}{q^3}+\cdots. \] Denote by $\mathbf U_q$ the set of corresponding sequences of all points in $\mathcal U_q$. It is well-known that the function $H: q\mapsto h(\mathbf U_q)$ is a Devil's staircase, where $h(\mathbf U_q)$ denotes the topological entropy of $\mathbf U_q$. In this paper we {give several characterizations of} the bifurcation set \[ \mathcal B:=\{q\in(1,M+1]: H(p)\ne H(q)\textrm{ for any }p\ne q\}. \] Note that $\mathcal B$ is contained in the set $\mathcal{U}^R$ of bases $q\in(1,M+1]$ such that $1\in\mathcal U_q$. By using a transversality technique we also calculate the Hausdorff dimension of the difference $\mathcal B\backslash\mathcal{U}^R$. Interestingly this quantity is always strictly between $0$ and $1$. When $M=1$ the Hausdorff dimension of $\mathcal B\backslash\mathcal{U}^R$ is $\frac{\log 2}{3\log λ^*}\approx 0.368699$, where $λ^*$ is the unique root in $(1, 2)$ of the equation $x^5-x^4-x^3-2x^2+x+1=0$.

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arXiv2018

Algebraic sums and products of univoque bases

Given $x\in(0, 1]$, let $\mathcal U(x)$ be the set of bases $q\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^\infty d_i/q^i$. Lü, Tan and Wu (2014) proved that $\mathcal U(x)$ is a Lebesgue null set of full Hausdorff dimension. In this paper, we show that the algebraic sum $\mathcal U(x)+λ\mathcal U(x)$ and product $\mathcal U(x)\cdot\mathcal U(x)^λ$ contain an interval for all $x\in(0, 1]$ and $λ\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters studied by the first author and Kalle (2017).

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