arXiv subjects

Derong Kong

Publications and source records attributed to Derong Kong.

At least 19 recordsLinked to original sources

arXiv2026

Phase transitions on periodic orbits in $\beta$-transformation with a hole at zero

Given $\beta\in(1,2]$, let $T_\beta: [0,1)\to[0,1);~x\mapsto\beta x\pmod 1$. For $m\in\mathbb N$ let \[ \tau_m(\beta):=\sup\left\{t\in[0,1): K_\beta(t)\textrm{ {contains a periodic orbit} of smallest period }m \right\}, \] where $K_\beta(t)=\{x\in[0,1): T_\beta^n(x)\notin(0,t)~\forall n\ge 0\}$ is the survivor set of the open dynamical system $(T_\beta, [0,1), H)$ with a hole $H=(0,t)$. In this paper we give a complete characterization of $\tau_m$, and show that $\tau_m$ is piecewise continuous with precisely $\psi(m)$ discontinuity points, where $\psi(m)$ is the number of bulbs of period $m$ in the Mandelbrot set. To describe the critical value function $\tau_m$ we construct a finite butterfly tree $\mathcal T_m$, from which we are able to determine the discontinuity points and the analytic formula of $\tau_m$ based on Farey words and substitution operators. As a by product, we characterize the extremal Lyndon words and extremal Perron words. Since we are working in the symbolic space, our result can be applied to study phase transitions for periodic orbits in topologically expansive Lorenz maps, doubling map with an asymmetric hole, intermediate $\beta$-transformations, unique expansions in double bases, and so on.

math.DS↗

arXiv2025

Luminance-Aware Statistical Quantization: Unsupervised Hierarchical Learning for Illumination Enhancement

Low-light image enhancement (LLIE) faces persistent challenges in balancing reconstruction fidelity with cross-scenario generalization. While existing methods predominantly focus on deterministic pixel-level mappings between paired low/normal-light images, they often neglect the continuous physical process of luminance transitions in real-world environments, leading to performance drop when normal-light references are unavailable. Inspired by empirical analysis of natural luminance dynamics revealing power-law distributed intensity transitions, this paper introduces Luminance-Aware Statistical Quantification (LASQ), a novel framework that reformulates LLIE as a statistical sampling process over hierarchical luminance distributions. Our LASQ re-conceptualizes luminance transition as a power-law distribution in intensity coordinate space that can be approximated by stratified power functions, therefore, replacing deterministic mappings with probabilistic sampling over continuous luminance layers. A diffusion forward process is designed to autonomously discover optimal transition paths between luminance layers, achieving unsupervised distribution emulation without normal-light references. In this way, it considerably improves the performance in practical situations, enabling more adaptable and versatile light restoration. This framework is also readily applicable to cases with normal-light references, where it achieves superior performance on domain-specific datasets alongside better generalization-ability across non-reference datasets.

cs.CV↗

arXiv2025

On the intersection of Cantor set with the unit circle and some sequences

For $\lambda\in(0,1/2)$ let $K_\lambda$ be the self-similar set in $\mathbb{R}$ generated by the iterated function system $\{f_0(x)=\lambda x, f_1(x)=\lambda x+1-\lambda \}$. In this paper, we investigate the intersection of the unit circle $\mathbb{S} \subset \mathbb{R}^2$ with the Cartesian product $K_{\lambda} \times K_{\lambda}$. We prove that for $\lambda \in(0, 2 - \sqrt{3}]$, the intersection is trivial, i.e., \[ \mathbb{S} \cap (K_{\lambda} \times K_{\lambda}) = \{(0,1), (1,0)\}. \] If $\lambda\in [0.330384,1/2)$, then the intersection $\mathbb{S} \cap (K_{\lambda} \times K_{\lambda})$ is non-trivial. In particular, if $\lambda\in [0.407493 , 1/2)$ the intersection $\mathbb{S} \cap (K_{\lambda} \times K_{\lambda})$ is of cardinality continuum. Furthermore, the bound $2 - \sqrt{3}$ is sharp: there exists a sequence $\{\lambda_n\}_{n \in \mathbb{N}}$ with $\lambda_n \searrow 2 - \sqrt{3}$ such that $\mathbb{S} \cap (K_{\lambda_n} \times K_{\lambda_n})$ is non-trivial for all $n\in\mathbb{N}$. This result provides a negative answer to a problem posed by Yu (2023). Our methods extend beyond the unit circle and remain effective for many nonlinear curves. By employing tools from number theory, including the quadratic reciprocity law, we analyze the intersection of Cantor sets with some sequences. A dichotomy is established in terms of the Legendre symbol associated with the digit set, revealing a fundamental arithmetic constraint governing such intersections.

math.CA↗

arXiv2025

Open dynamical systems with a moving hole

Given an integer $b\ge 3$, let $T_b: [0,1)\to [0,1); x\mapsto bx\pmod 1$ be the expanding map on the unit circle. For any $m\in\mathbb{N}$ and $\omega=\omega^0\omega^1\ldots\in(\left\{0,1,\ldots,b-1\right\}^m)^\mathbb{N_0}$ let \[ K^\omega=\left\{x\in[0,1): T_b^n(x)\notin I_{\omega^n}~\forall n\geq 0\right\},\] where $I_{\omega^n}$ is the $b$-adic basic interval generated by $\omega^n$. Then $K^\omega$ is called the survivor set of the open dynamical system $([0,1),T_b,I_\omega)$ with respect to the sequence of holes $I_\omega=\left\{I_{\omega^n}: n\geq 0\right\}$. We show that the Hausdorff and lower box dimensions of $K^\omega$ always conincide, and the packing and upper box dimensions of $K^\omega$ also coincide. Moreover, we give sharp lower and upper bounds for the dimensions of $K^\omega$, which can be calculated explicitly. For any admissible $\alpha\leq \beta$ there exist infinitely many $\omega$ such that $\dim_H K^\omega=\alpha$ and $\dim_P K^\omega=\beta$. As applications we study badly approximable numbers in Diophantine approximation. For an arbitrary sequence of balls $\left\{B_n\right\}$, let $K\left(\left\{B_n\right\}\right)$ be the set of $x\in[0,1)$ such that $T_b^n(x)\notin B_n$ for all but finitely many $n\geq 0$. Assuming $\lim_{n\to\infty}\operatorname{diam} \left(B_n\right)$ exists, we show that $\dim_H K\left(\left\{B_n\right\}\right)=1$ if and only if $\lim_{n\to\infty}\operatorname{diam} \left(B_n\right)=0$. For any positive function $\phi$ on $\mathbb{N}$, let $E\left(\phi\right)$ be the set of $x\in[0,1)$ satisfying $|T_b^n (x)-x|\geq \phi(n)$ for all but finitely many $n$. If $\lim_{n\to\infty}\phi(n)$ exists, then $\dim_H E(\phi)=1$ if and only if $\lim_{n\to\infty}\phi(n)=0$. Our results can be applied to study joint spectral radius of matrices. We show that the finiteness property for the joint spectral radius of associated adjacency matrices holds true.

math.DS↗

arXiv2025

Rational points in Cantor sets and spectral eigenvalue problem for self-similar spectral measures

Given $q\in \mathbb{N}_{\ge 3}$ and a finite set $A\subset\mathbb{Q}$, let $$K(q,A)= \bigg\{\sum_{i=1}^{\infty} \frac{a_i}{q^{i}}:a_i \in A ~\forall i\in \mathbb{N} \bigg\}.$$ For $p\in\mathbb{N}_{\ge 2}$ let $D_p\subset\mathbb{R}$ be the set of all rational numbers having a finite $p$-ary expansion. We show in this paper that for $p \in \mathbb{N}_{\ge 2}$ with $\gcd(p,q)=1$, the intersection $D_p\cap K(q, A)$ is a finite set if and only if $\dim_H K(q, A)<1$, which is also equivalent to the fact that the set $K(q, A)$ has no interiors. We apply this result to study the spectral eigenvalue problem. For a Borel probability measure $\mu$ on $\mathbb{R}$, a real number $t\in \mathbb{R}$ is called a spectral eigenvalue of $\mu$ if both $E(\Lambda) =\big\{ e^{2 \pi \mathrm{i} \lambda x}: \lambda \in \Lambda \big\}$ and $E(t\Lambda) = \big\{ e^{2 \pi \mathrm{i} t\lambda x}: \lambda \in \Lambda \big\}$ are orthonormal bases in $L^2(\mu)$ for some $\Lambda \subset \mathbb{R}$. For any self-similar spectral measure generated by a Hadamard triple, we provide a class of spectral eigenvalues which is dense in $[0,+\infty)$, and show that every eigen-subspace associated with these spectral eigenvalues is infinite.

math.CA↗

arXiv2025

Phase transitions for unique codings of fat Sierpinski gaskets with multiple digits

Given an integer $M\ge 1$ and $\beta\in(1, M+1)$, let $S_{\beta, M}$ be the fat Sierpinski gasket in $\mathbb R^2$ generated by the iterated function system $\left\{f_d(x)=\frac{x+d}{\beta}: d\in\Omega_M\right\}$, where $\Omega_M=\{(i,j)\in\mathbb Z_{\ge 0}^2: i+j\le M\}$. Then each $x\in S_{\beta, M}$ may be represented as a series $x=\sum_{i=1}^\infty\frac{d_i}{\beta^i}=:\Pi_\beta((d_i))$, and the infinite sequence $(d_i)\in\Omega_M^{\mathbb N}$ is called a \emph{coding} of $x$. Since $\beta \beta_c(M)$ then $U_{\beta, M}$ has positive Hausdorff dimension. Our results can also be applied to the intrinsic univoque set $\widetilde{U}_{\beta, M}$. Moreover, we show that the first critical base $\beta_G(M)$ is a Perron number, while the second critical base $\beta_c(M)$ is a transcendental number.

math.DS↗

arXiv2024

The $\beta$-transformation with a hole at $0$: the general case

Given $\beta>1$, let $T_\beta$ be the $\beta$-transformation on the unit circle $[0,1)$, defined by $T_\beta(x)=\beta x-\lfloor \beta x\rfloor$. For each $t\in[0,1)$ let $K_\beta(t)$ be the survivor set consisting of all $x\in[0,1)$ whose orbit $\{T^n_\beta(x): n\ge 0\}$ never hits the interval $[0,t)$. Kalle et al.~[{\em Ergodic Theory Dynam. Systems} {\bf 40} (2020), no.~9, 2482--2514] considered the case $\beta\in(1,2]$. They studied the set-valued bifurcation set $\mathscr{E}_\beta:=\{t\in[0,1): K_\beta(t')\ne K_\beta(t)~\forall t'>t\}$ and proved that the Hausdorff dimension function $t\mapsto\dim_H K_\beta(t)$ is a non-increasing Devil's staircase. In a previous paper [{\em Ergodic Theory Dynam. Systems} {\bf 43} (2023), no.~6, 1785--1828] we determined, for all $\beta\in(1,2]$, the critical value $\tau(\beta):=\min\{t>0: \eta_\beta(t)=0\}$. The purpose of the present article is to extend these results to all $\beta>1$. In addition to calculating $\tau(\beta)$, we show that (i) the function $\tau: \beta\mapsto\tau(\beta)$ is left continuous on $(1,\infty)$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau$ has no downward jumps; and (iii) there exists an open set $O\subset(1,\infty)$, whose complement $(1,\infty)\backslash O$ has zero Hausdorff dimension, such that $\tau$ is real-analytic, strictly convex and strictly decreasing on each connected component of $O$. We also prove several topological properties of the bifurcation set $\mathscr{E}_\beta$. The key to extending the results from $\beta\in(1,2]$ to all $\beta>1$ is an appropriate generalization of the Farey words that are used to parametrize the connected components of the set $O$. Some of the original proofs from the above-mentioned papers are simplified.

math.DS↗

arXiv2024

Fractal Sumset Properties

In this paper we introduce two notions of fractal sumset properties. A compact set $K\subset\mathbb{R}^d$ is said to have the Hausdorff sumset property (HSP) if for any $\ell\in\mathbb{N}_{\ge 2}$ there exist compact sets $K_1, K_2,\ldots, K_\ell$ such that $K_1+K_2+\cdots+K_\ell\subset K$ and $\dim_H K_i=\dim_H K$ for all $1\le i\le \ell$. Analogously, if we replace the Hausdorff dimension by the packing dimension in the definition of HSP, then the compact set $K\subset\mathbb{R}^d$ is said to have the packing sumset property (PSP). We show that the HSP fails for certain homogeneous self-similar sets satisfying the strong separation condition, while the PSP holds for all homogeneous self-similar sets in $\mathbb{R}^d$.

math.CA↗

arXiv2024

Rational Points in Translations of The Cantor Set

Given two coprime integers $p\ge 2$ and $q \ge 3$, let $D_p\subset[0,1)$ consist of all rational numbers which have a finite $p$-ary expansion, and let $$ K(q, \mathcal{A})=\bigg\{ \sum_{i=1}^\infty \frac{d_i}{q^i}: d_i\in \mathcal{A}~ \forall i\in\mathbb{N} \bigg\}, $$ where $\mathcal{A} \subset \{0,1,\ldots, q-1\}$ with cardinality $1<\#\mathcal{A}< q$. In 2021 Schleischitz showed that $\#(D_p\cap K(q,\mathcal{A}))<+\infty$. In this paper we show that for any $r\in\mathbb{Q}$ and for any $α\in\mathbb{R}$, $$ \#\big((r D_p+α)\cap K(q,\mathcal{A})\big)<+\infty. $$

math.NT↗

arXiv2024

On a class of self-similar sets which contain finitely many common points

For $λ\in(0,1/2]$ let $K_λ\subset\mathbb{R}$ be a self-similar set generated by the iterated function system $\{λx, λx+1-λ\}$. Given $x\in(0,1/2)$, let $Λ(x)$ be the set of $λ\in(0,1/2]$ such that $x\in K_λ$. In this paper we show that $Λ(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\ldots, y_p\in(0,1/2)$ there exists a full Hausdorff dimensional set of $λ\in(0,1/2]$ such that $y_1,\ldots, y_p \in K_λ$.

math.DS↗

arXiv2024

Projections of four corner Cantor set: total self-similarity, spectrum and unique codings

Given $ρ\in (0,1/4]$, the four corner Cantor set $E\subset \mathbb{R}^{2}$ is a self-similar set generated by the iterated function system \[ \left\{(ρx, ρy), \quad(ρx, ρy+1-ρ),\quad (ρx+1-ρ, ρy),\quad(ρx+1-ρ,ρy+1-ρ)\right\}. \] For $θ\in[0,π)$ let $E_θ$ be the orthogonal projection of $E$ onto a line with an angle $θ$ to the $x$-axis. In this paper we give a complete characterization on which the projection $E_θ$ is totally self-similar. We also study the spectrum of $E_θ$, which turns out that the spectrum of $E_θ$ achieves its maximum value if and only if $E_θ$ is totally self-similar. Furthermore, when $E_θ$ is totally self-similar, we calculate its Hausdorff dimension and study the subset $U_θ$ which consists of all $x\in E_θ$ having a unique coding. In particular, we show that $\dim_H U_θ=\dim_H E_θ$ for Lebesgue almost every $θ\in[0,π)$. Finally, for $ρ=1/4$ we describe the distribution of $θ$ in which $E_θ$ contains an interval. It turns out that the possibility for $E_θ$ to contain an interval is smaller than that for $E_θ$ to have an exact overlap.

math.DS↗

arXiv2024

On the strong separation condition for self-similar iterated function systems with random translations

Given a self-similar iterated function system $\Phi=\{ \phi_i(x)=\rho_i O_i x+t_i \}_{i=1}^m$ acting on $\mathbb{R}^d$, we can generate a parameterised family of iterated function systems by replacing each $t_i$ with a random vector in $\mathbb{R}^d$. In this paper we study whether a Lebesgue typical member of this family will satisfy the strong separation condition. Our main results show that if the similarity dimension of $\Phi$ is sufficiently small, then a Lebesgue typical member of this family will satisfy the strong separation condition.

math.DS↗

arXiv2023

Periodic unique codings of fat Sierpinski gasket

For $β>1$ let $S_β$ be the Sierpinski gasket generated by the iterated function system \[\left\{f_{α_0}(x,y)=\Big(\frac{x}β,\frac{y}β\Big), \quad f_{α_1}(x,y)=\Big(\frac{x+1}β, \frac{y}β\Big), \quad f_{α_2}(x,y)=\Big(\frac{x}β, \frac{y+1}β\Big)\right\}.\] If $β\in(1,2]$, then the overlap region $O_β:=\bigcup_{i\ne j}f_{α_i}(Δ_β)\cap f_{α_j}(Δ_β)$ is nonempty, where $Δ_β$ is the convex hull of $S_β$. In this paper we study the periodic codings of the univoque set \[ \mathbf U_β:=\left\{(d_i)_{i=1}^\infty\in\{(0,0), (1,0), (0,1)\}^\mathbb N: \sum_{i=1}^\infty d_{n+i}β^{-i}\in S_β\setminus O_β~\forall n\ge 0\right\}. \] More precisely, we determine for each $k\in\mathbb N$ the smallest base $β_k\in(1,2]$ such that for any $β>β_k$ the set $\mathbf U_β$ contains a sequence of smallest period $k$. We show that each $β_k$ is a Perron number, and the sequence $(β_k)$ has infinitely many accumulation points. Furthermore, we show that $β_{3k}>β_{3\ell}$ if and only if $k$ is larger than $\ell$ in the Sharkovskii ordering; and the sequences $ (β_{3\ell+1}), (β_{3\ell+2})$ decreasingly converge to the same limit point $β_a\approx 1.55898$, respectively. In particular, we find that $β_{6m+4}=β_{3m+2}$ for all $m\ge 0$. Consequently, we prove that if $\mathbf U_β$ contains a sequence of smallest period $2$ or $4$, then $\mathbf U_β$ contains a sequence of smallest period $k$ for any $k\in\mathbb N$.

math.DS↗

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.

math.DS↗

arXiv2023

Entropy plateaus, transitivity and bifurcation sets for the $\beta$-transformation with a hole at $0$

Given $\beta>1$, let $T_\beta$ be the $\beta$-transformation on the unit circle $[0,1)$ such that $T_\beta(x)=\beta x\pmod 1$. For each $t\in[0,1)$ let $K_\beta(t)$ be the survivor set consisting of all $x\in[0,1)$ whose orbit $\{T^n_\beta(x): n\ge 0\}$ never enters the interval $[0,t)$. Letting $\mathscr{E}_\beta$ denote the bifurcation set of the set-valued map $t\mapsto K_\beta(t)$, Kalle et al. [Ergodic Theory Dynam. Systems, 40 (9): 2482--2514, 2020] conjectured that \[ \dim_H\big(\mathscr{E}_\beta\cap[t,1]\big)=\dim_H K_\beta(t) \qquad \forall\,t\in(0,1). \] The main purpose of this article is to prove this conjecture. We do so by investigating dynamical properties of the symbolic equivalent of the survivor set $K_\beta(t)$, in particular its entropy and topological transitivity. In addition, we compare $\mathscr{E}_\beta$ with the bifurcation set $\mathscr{B}_\beta$ of the map $t\mapsto \dim_H K_\beta(t)$ (which is a decreasing devil's staircase by a theorem of Kalle et al.), and show that, for Lebesgue-almost every $\beta>1$, the difference $\mathscr{E}_\beta\backslash\mathscr{B}_\beta$ has positive Hausdorff dimension, but for every $k\in\{0,1,2,\dots\}\cup\{\aleph_0\}$, there are infinitely many values of $\beta$ such that the cardinality of $\mathscr{E}_\beta\backslash\mathscr{B}_\beta$ is exactly $k$. For a countable but dense subset of $\beta$'s, we also determine the intervals of constancy of the function $t\mapsto \dim_H K_\beta(t)$. Some connections with other topics in dynamics, such as kneading invariants of Lorenz maps and the doubling map with an arbitrary hole, are also discussed.

math.DS↗

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.

math.DS↗

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 $q$.

math.DS↗

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.

math.DS↗