Hausdorff Dimension Research Articles

Birkhoff spectrum for diagonally self-affine sets and digit frequencies for GLS systems with redundancy

In this article, we calculate the Birkhoff spectrum in terms of the Hausdorff dimension of level sets for Birkhoff averages of continuous potentials for a certain family of diagonally affine iterated function systems. Also, we study Besicovitch–Eggleston sets for finite generalized Lüroth series number systems with redundancy. The redundancy refers to the fact that each number $x \in [0,1]$ has uncountably many expansions in the system. We determine the Hausdorff dimension of digit frequency sets for such expansions along fibres.

ON ARITHMETIC PROGRESSIONS IN THE DIGITS OF ENGEL EXPANSIONS

Denote by [Formula: see text] and [Formula: see text] the sets of real numbers whose Engel expansion digits contain arbitrarily long arithmetic progressions and arbitrarily long arithmetic progressions with arbitrary common difference, respectively. In this paper, the sizes of [Formula: see text] and [Formula: see text] are investigated from the measure-theoretical, fractal and topological viewpoints. We prove that [Formula: see text] and [Formula: see text] are of Lebesgue measure zero but have full Hausdorff dimension. Moreover, [Formula: see text] is of first category while [Formula: see text] is residual.

Generalized Dimensions of Self-Affine Sets with Overlaps

Two decades ago, Ngai and Wang introduced a well-known finite type condition (FTC) on the self-similar iterated function system (IFS) with overlaps and used it to calculate the Hausdorff dimension of self-similar sets. In this paper, inspired by Ngai and Wang’s idea, we define a new FTC on self-affine IFS and obtain an analogous formula on the generalized dimensions of self-affine sets. The generalized dimensions raised by He and Lau are used to estimate the Hausdorff dimension of self-affine sets.

The Hausdorff Dimension of the Sets of Irrationals with Prescribed Relative Growth Rates

The Hausdorff Dimension of the Sets of Irrationals with Prescribed Relative Growth Rates

Sharp Hausdorff content estimates for accessible boundaries of domains in metric measure spaces of controlled geometry

Sharp Hausdorff content estimates for accessible boundaries of domains in metric measure spaces of controlled geometry

Exceptional sets to Shallit's law of leap years in Pierce expansions

In his 1994 work, Shallit introduced a rule for determining leap years that generalizes both the historically used Julian calendar and the contemporary Gregorian calendar. This rule depends on a so-called intercalation sequence. According to what we term Shallit's law of leap years, almost every point of the interval [0,1] with respect to the Lebesgue measure has the same limsup and liminf, respectively, of a quotient defined in terms of the number of leap years determined by the rule using the Pierce expansion digit sequence as an intercalation sequence. In this paper, we show that the set of exceptions to this law is dense and has full Hausdorff dimension in [0,1], and that the exceptional set intersected with any non-empty open subset of [0,1] has full Hausdorff dimension in [0,1]. As a more general result, we establish that for certain subsets of [0,1] concerning the limiting behavior of Pierce expansion digits, intersecting with a non-empty open subset of [0,1] preserves the Hausdorff dimension.

Amenable metric mean dimension and amenable mean Hausdorff dimension of product sets and metric varying

Metric mean dimension and mean Hausdorff dimension depend on metrics. In this paper, we investigate the continuity of the metric mean dimension and mean Hausdorff dimension concerning the metrics for amenable group actions, which extends recent results by Muentes, Becker, Baraviera et al.. Moreover, we prove the product formulas for the mean Hausdorff dimension and the metric mean dimension for amenable group actions.

Metrical properties of products formed by consecutive partial quotients in Lüroth expansions

For x∈(0,1], let [d1(x),…,dk(x),…] be its Lüroth expansion, and {pk(x)qk(x)}k≥1 be its convergents. Brown-Sarre et al. considered the metrical properties of products derived from sequential partial quotients raised to different powers. More precisely, for any t=(t0,…,tm)∈R+m+1, they examined the metrical characteristics of the set Et(ψ)={x∈(0,1]:dkt0(x)dk+1t1(x)⋯dk+mtm(x)≥ψ(k)for i.mk},where m represents a positive integer, ψ:N→(1,∞) is a function. In this paper, we will proceed with the study of these issues. Namely, the primary aim of this research is to explore the metrical properties of the following sets Dt(ψ)={x∈(0,1]:dkt0(x)dk+1t1(x)⋯dk+mtm(x)≥ψ(k),k≥1}and Ft(ψ)={x∈(0,1]:dkt0(x)dk+1t1(x)⋯dk+mtm(x)≥ψ(qk(x))for i.m.k}.Moreover, we also give due consideration to the Hausdorff dimension of the exceptional set associated with the convergence exponent cm(x)=inf{l≥0:∑k=1∞(dk(x)dk+1(x)⋯dk+m(x))−l<∞}.

Hausdorff dimension of large quadratic Weyl sums

Hausdorff dimension of large quadratic Weyl sums

Metrical properties of exponentially growing partial quotients

Abstract A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers especially when their partial quotients exhibit specific growth rates. For any positive function Φ, the Wang–Wu theorem (2008) comprehensively describes the Hausdorff dimension of the set E 1 ( Φ ) : = { x ∈ [ 0 , 1 ) : a n ( x ) ≥ Φ ( n ) for infinitely many n ∈ N } . \mathcal{E}_{1}(\Phi):=\{x\in[0,1):a_{n}(x)\geq\Phi(n)\ \text{for infinitely many}\ n\in\mathbb{N}\}. Various generalisations of this set exist, such as substituting one partial quotient with the product of consecutive partial quotients in the aforementioned set which has connections with the improvements to Dirichlet’s theorem, and many other sets of similar nature. Establishing the upper bound of the Hausdorff dimension of such sets is significantly easier than proving the lower bound. In this paper, we present a unified approach to get an optimal lower bound for many known setups, including results by Wang–Wu [Adv. Math. (2008)], Huang–Wu–Xu [Israel J. Math. (2020)], Tan–Zhou [Nonlinearity (2023)], and several others. We also provide a new theorem derived as an application of our main result. We do this by finding an exact Hausdorff dimension of the set S m ⁢ ( A 0 , … , A m − 1 ) ⁢ = def ⁢ { x ∈ [ 0 , 1 ) : c i ⁢ A i n ≤ a n + i ⁢ ( x ) &lt; 2 ⁢ c i ⁢ A i n , 0 ≤ i ≤ m − 1 , for infinitely many ⁢ n ∈ N } , S_{m}(A_{0},\ldots,A_{m-1})\overset{\mathrm{def}}{=}\{x\in[0,1):c_{i}A_{i}^{n}\leq a_{n+i}(x)&lt;2c_{i}A_{i}^{n},\,0\leq i\leq m-1,\,\text{for infinitely many}\ n\in\mathbb{N}\}, where each partial quotient grows exponentially and the base is given by a parameter A i &gt; 1 A_{i}&gt;1 . For proper choices of A i A_{i} , this set serves as a subset for sets under consideration, providing an optimal lower bound of Hausdorff dimension in all of them. The crux of the proof lies in introducing multiple probability measures consistently distributed over the Cantor-type subset of S m ⁢ ( A 0 , … , A m − 1 ) S_{m}(A_{0},\ldots,A_{m-1}) .

Dichotomy Law for a Modified Shrinking Target Problem in Beta Dynamical System

Let φ:N→(0,1] be a positive function. We consider the size of the set Ef(φ):={β&gt;1:|Tβn(x)−f(β)|&lt;φ(n)i.o.n}, where “i.o.n” stands for “infinitely often”, and f:(1,∞)→[0,1] is a Lipschitz function. For any x∈(0,1], it is proved that the Hausdorff measure of Ef(φ) fulfill a dichotomy law according to lim supn→∞logφ(n)n=−∞ or not, where Tβ is the β-transformation. In ergodic theory, the phenomenon of shrinking targets is crucial for understanding the long-term behavior of systems. By studying the shrinking target problem of the β dynamical system, we can reveal the relationship between randomness and determinism, which is significant for constructing more complex mathematical models. Moreover, there is a close connection between the β transformation and number theory. Investigating the contraction target problem helps uncover new properties and patterns in number theory, advancing the development of this field. In this work, we establish a significant relationship between the decay rate of the positive function φ(n) and the structural properties of the set Ef(φ). Specifically, we show that: The Hausdorff dimension of Ef(φ) either vanishes or is positive based on the behavior of φ(n) as n approaches infinity. The establishment of this dichotomy can help us more effectively understand the geometric characteristics and dynamical behavior of the system, thereby aiding our acceptance and comprehension of complex theories. Researching this shrinking target problem can help us uncover new properties in number theory, leading to a better understanding of the structure of numbers and promoting the development of related fields in number theory.

Hausdorff Dimension of a Class of Colored Substitution Networks

Hausdorff Dimension of a Class of Colored Substitution Networks

Shrinking parallelepiped targets for $\beta $ -dynamical systems

Abstract For $ \beta&gt;1 $ , let $ T_\beta $ be the $\beta $ -transformation on $ [0,1) $ . Let $ \beta _1,\ldots ,\beta _d&gt;1 $ and let $ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in $ [0,1)^d $ . Define $$ \begin{align*}W(\mathcal P)=\{\mathbf{x}\in[0,1)^d:(T_{\beta_1}\times\cdots \times T_{\beta_d})^n(\mathbf{x})\in P_n\text{ infinitely often}\}.\end{align*} $$ When each $ P_n $ is a hyperrectangle with sides parallel to the axes, the ‘rectangle to rectangle’ mass transference principle by Wang and Wu [Mass transference principle from rectangles to rectangles in Diophantine approximation. Math. Ann.381 (2021) 243–317] is usually employed to derive the lower bound for $\dim _{\mathrm {H}} W(\mathcal P)$ , where $\dim _{\mathrm {H}}$ denotes the Hausdorff dimension. However, in the case where $ P_n $ is still a hyperrectangle but with rotation, this principle, while still applicable, often fails to yield the desired lower bound. In this paper, we determine the optimal cover of parallelepipeds, thereby obtaining $\dim _{\mathrm {H}} W(\mathcal P)$ . We also provide several examples to illustrate how the rotations of hyperrectangles affect $\dim _{\mathrm {H}} W(\mathcal P)$ .

Bowen's Formula for a Dynamical Solenoid.

More than 50 years ago, Rufus Bowen noticed a natural relation between the ergodic theory and the dimension theory of dynamical systems. He proved a formula, known today as the Bowen's formula, that relates the Hausdorff dimension of a conformal repeller to the zero of a pressure function defined by a single conformal map. In this paper, we extend the result of Bowen to a sequence of conformal maps. We present a dynamical solenoid, i.e., a generalized dynamical system obtained by backward compositions of a sequence of continuous surjections (fn:X→X)n∈N defined on a compact metric space (X,d). Under mild assumptions, we provide a self-contained proof that Bowen's formula holds for dynamical conformal solenoids. As a corollary, we obtain that the Bowen's formula holds for a conformal surjection f:X→X of a compact.

Entanglement fractalization

We numerically explore the interplay of fractal geometry and quantum entanglement by analyzing the von Neumann entropy (known as entanglement entropy) and the entanglement contour in the scaling limit. Adopting quadratic fermionic models on Sierpinski carpet, we uncover intriguing findings. For gapless ground states exhibiting a finite density of states at the chemical potential, we reveal a super-area law characterized by the presence of a logarithmic correction for area law in the scaling of entanglement entropy. This extends the well-established super-area law observed on translationally invariant Euclidean lattices where the Gioev-Klich-Widom conjecture regarding the asymptotic behavior of Toeplitz matrices holds significant influence. Furthermore, different from the fractal structure of the lattice, we observe the emergence of a self-similar and universal pattern termed an “entanglement fractal” in the entanglement contour data as we approach the scaling limit. Remarkably, this pattern bears resemblance to intricate Chinese paper-cutting designs. We provide general rules to artificially generate this fractal, offering insights into the universal scaling of entanglement entropy. Meanwhile, as the direct consequence of the entanglement fractal and beyond a single scaling behavior of entanglement contour in translation-invariant systems, we identify two distinct scaling behaviors in the entanglement contour of fractal systems. For gapped ground states, we observe that the entanglement entropy adheres to a generalized area law, with its dependence on the Hausdorff dimension of the boundary between complementary subsystems. Published by the American Physical Society 2024

On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets

On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets

On an abrasion-motivated fractal model

Abstract In this paper, we consider a fractal model motivated by the abrasion of convex polyhedra, where the abrasion is realised by chipping small neighbourhoods of vertices. After providing a formal description of the successive chippings, we show that the net of edges converge to a compact limit set under mild assumptions. Furthermore, we study the upper box-counting dimension and the Hausdorff dimension of the limiting object of the net of edges after infinitely many chipping.

Equilibrium measures on Julia sets of random quadratic polynomials

Abstract We consider sequences of compositions of quadratic polynomials f c n ( z ) = z 2 + c n . For such sequences one can naturally generalise the definitions of the Julia set and basin of infinity from the autonomous case. In this setting the Julia set depends on a sequence ω = ( c 0 , c 1 , … ) . We study the equilibrium (harmonic) measure on such Julia sets. In particular, we calculate the Hausdorff dimension of the equilibrium measure and study its dependence on the ‘scale of randomnes’.

An upper bound of the Hausdorff dimension of singular vectors on affine subspaces

In Diophantine approximation, the notion of singular vectors was introduced by Khintchine in the 1920’s. We study the set of singular vectors on an affine subspace of R n \mathbb {R}^n . We give an upper bound of its Hausdorff dimension in terms of the Diophantine exponent of the parameter of the affine subspace.

Dimension estimates for badly approximable affine forms

Abstract For given $\epsilon&gt;0$ and $b\in \mathbb {R}^m$ , we say that a real $m\times n$ matrix A is $\epsilon $ -badly approximable for the target b if $$ \begin{align*}\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b\rangle^m \geq \epsilon,\end{align*} $$ where $\langle \cdot \rangle $ denotes the distance from the nearest integral vector. In this article, we obtain upper bounds for the Hausdorff dimensions of the set of $\epsilon $ -badly approximable matrices for fixed target b and the set of $\epsilon $ -badly approximable targets for fixed matrix A. Moreover, we give a Diophantine condition of A equivalent to the full Hausdorff dimension of the set of $\epsilon $ -badly approximable targets for fixed A. The upper bounds are established by effectivizing entropy rigidity in homogeneous dynamics, which is of independent interest. For the A-fixed case, our method also works for the weighted setting where the supremum norms are replaced by certain weighted quasinorms.