Hausdorff Dimension Research Articles

Quantitative recurrence properties in Besicovitch-Eggleston sets

Let Tx=bx(mod1) and Ax(i,b,N) denote the number of occurrences of the digit i in the first N digits of b-adic expansion of x. We study the quantitative recurrence properties in Besicovitch-Eggleston sets, which is defined asRb,ρ(ψ)={x∈[0,1):|Tnx−x|<ψ(n)for infinitely manyn∈NandlimN→∞⁡Ax(i,b,N)N=pifor anyi∈{0,⋯,b−1}}, where ρ=(p0,⋯,pb−1) is a probability vector and ψ: N→(0,1] is a positive function. In this paper, the Hausdorff dimension of the set Rb,ρ(ψ) is determined.

Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces

We prove that for a quasiself-similar and arcwise connected compact metric space all three known versions of the conformal dimension coincide: the conformal Hausdorff dimension, conformal Assouad dimension and Ahlfors regular conformal dimension. This answers a question posed by Murugan. Quasisimilar spaces include all approximately self-similar spaces. As an example, the standard Sierpiński carpet is quasiself-similar and thus the three notions of conformal dimension coincide for it. We also give the equality of the three dimensions for combinatorially p-Loewner (CLP) spaces. Both proofs involve using a new notion of combinatorial modulus, which lies between two notions of modulus that have appeared in the literature. The first of these is the modulus studied by Pansu and Tyson, which uses a Carathéodory construction. The second is the one used by Keith and Laakso (and later modified and used by Bourdon, Kleiner, Carrasco-Piaggio, Murugan and Shanmugalingam). By combining these approaches, we gain the flexibility of giving upper bounds for the new modulus from the Pansu–Tyson approach, and the ability of getting lower bounds using the Keith–Laakso approach. Additionally the new modulus can be iterated in self-similar spaces, which is a crucial, and novel, step in our argument.

On the Local Fourier Uniformity Problem for Small Sets

Abstract We consider vanishing properties of exponential sums of the Liouville function $\boldsymbol{\lambda }$ of the form $$ \begin{align*} &amp; \lim_{H\to\infty}\limsup_{X\to\infty}\frac{1}{\log X}\sum_{m\leq X}\frac{1}{m}\sup_{\alpha\in C}\bigg|\frac{1}{H}\sum_{h\leq H}\boldsymbol{\lambda}(m+h)e^{2\pi ih\alpha}\bigg|=0, \end{align*} $$ where $C\subset{{\mathbb{T}}}$. The case $C={{\mathbb{T}}}$ corresponds to the local $1$-Fourier uniformity conjecture of Tao, a central open problem in the study of multiplicative functions with far-reaching number-theoretic applications. We show that the above holds for any closed set $C\subset{{\mathbb{T}}}$ of zero Lebesgue measure. Moreover, we prove that extending this to any set $C$ with non-empty interior is equivalent to the $C={{\mathbb{T}}}$ case, which shows that our results are essentially optimal without resolving the full conjecture. We also consider higher-order variants. We prove that if the linear phase $e^{2\pi ih\alpha }$ is replaced by a polynomial phase $e^{2\pi ih^{t}\alpha }$ for $t\geq 2$ then the statement remains true for any set $C$ of upper box-counting dimension $&amp;lt; 1/t$. The statement also remains true if the supremum over linear phases is replaced with a supremum over all nilsequences coming form a compact countable ergodic subsets of any $t$-step nilpotent Lie group. Furthermore, we discuss the unweighted version of the local $1$-Fourier uniformity problem, showing its validity for a class of “rigid” sets (of full Hausdorff dimension) and proving a density result for all closed subsets of zero Lebesgue measure.

The principle of cascadic prosilience—Assessment of territorial recovery strategies after disrupted regional infrastructure systems in Iraq

This paper presents an assessment approach to fostering socioeconomic re-development and resilience in Iraqi regions emerging from the destruction and instability, in the aftermath of the war conflict in Iraq. Focusing on the intricate interplay of logistics infrastructure and economic recovery, the present study proposes a novel framework that integrates general resilience insights, data analytics, infrastructure systems, and decision support from Data Envelopment Analysis (DEA). We draw inspiration also from historical cases on “creative destruction” or “Blessing in Disguise” (BiD) phenomena, like the post-WWII reconstruction of Rotterdam, so as to develop the notion of stepwise or cascadic prosilience, analyzing how innovative logistics systems may in various stages contribute to economic rejuvenation. Our approach recognizes the multifaceted nature of regional resilience capacity, encompassing both static (conserving resources, rerouting, etc.) and dynamic (accelerating recovery through innovative strategies) dimensions. The logistics aspect spans both the supply side (new infrastructure, ICT facilities) and the demand side (changing transportation flows and product demands), culminating in an integrated perspective for sustainable growth of Iraqi regions. In our study, we explore several forward-looking strategic future options (scenarios) for recovery and reconstruction policy factors in the context of regional development in Iraq, regarding them as crucial strategic elements for effective post-conflict rebuilding and regeneration. Given that such assets and infrastructures typically extend beyond a single city or area, their geographic scope is broader, calling for a multi-region approach. By leveraging the extended DEA approach by an incorporation of a super-efficiency (SE) DEA approach so as to better discriminate among efficient Decision-Making Units (DMUs)—in this case, regions in Iraq—our research aims to present actionable and effective insights for infrastructure investment strategies at regional-governorate scale in Iraq, that optimize efficiency, sustainability and resilience. This approach may ultimately foster prosperous and stable post-conflict regional economies that display—by means of a cascadic change—a new balanced prosilient future.

A remark on the set of exactly approximable vectors in the simultaneous case

We compute the Hausdorff dimension of the set of ψ \psi -exactly approximable vectors, in the simultaneous case, in dimension strictly larger than 2 2 and for approximating functions ψ \psi with order at infinity less than or equal to − 2 -2 . Our method relies on the analogous result in dimension 1 1 , proved by Yann Bugeaud and Carlos Moreira, and a version of Jarník’s theorem on fibres.

Globally Exponentially Attracting Sets and Heteroclinic Orbits Revealed

Motivated by the open problems on the global dynamics of the generalized four-dimensional Lorenz-like system, this paper deals with the existence of globally exponentially attracting sets and heteroclinic orbits by constructing a series of Lyapunov functions. Specifically, not only is a family of mathematical expressions of globally exponentially attracting sets derived, but the existence of a pair of orbits heteroclinic to S0 and S± is also proven with the aid of a Lyapunov function and the definitions of both the α-limit set and ω-limit set. Moreover, numerical examples are used to justify the theoretical analysis. Since the obtained results improve and complement the existing ones, they may provide support in chaos control, chaos synchronization, the Hausdorff and Lyapunov dimensions of strange attractors, etc.

The effects of sensorial and mobility frailty on the overall and domain-specific cognition performance of Chinese community-dwelling older adults.

This study aimed to investigate the different impacts of sensorial and mobility frailty on overall and domain-specific cognitive function. Further, the independent associations between other intricate capacity (IC) dimensions, including vitality and psychological dimensions, and overall and domain-specific cognitive function were investigated. A total of 429 participants (mean age, 72.91 ± 7.014 years; 57.30% female) underwent IC capacity assessment. Other covariates, such as demographics, health-related variables were also assessed. Overall or domain-specific cognitive impairment was used as a dependent variable in logistic regression analyses adjusted for demographic, health-related, and psychosocial confounders. After adjustment for demographic, health-related, and psychosocial confounders, individuals with sensorial frailty (odds ratio [OR] = 0.435; 95% confidence interval [CI] = 0.236-0.801; P = .008) had a significantly lower risk of mild cognitive impairment (MCI), marginally low delayed memory impairment (OR = 0.601, 95% CI = 0.347-1.040; P = .069), and language impairment (OR = 0.534, 95% CI = 0.305-0.936; OR = 0.318, P = .029; OR = 0.318,95% CI = 0.173-0.586; P < .001) by Boston naming and animal fluency tests than did those with both sensorial and mobility frailty or mobility frailty only. Depressive symptoms had a significant negative influence on executive function. Cardiovascular disease and non-skin malignancy were independent determinants of MCI, and diabetes mellitus was independently associated with processing speed, attention, and executive function. Sensorial and mobility frailty were independent risk factors for cognitive impairment. Mobility frailty had a greater negative influence on the overall cognitive function and memory and language function than did sensorial frailty. The reserve decline in the psychological dimension of IC and chronic diseases also had a significant adverse influence on overall and domain-specific cognition function.

Role identities of emergency medical services personnel and their associations with intention to leave the profession

BackgroundThe scope of emergency medical services (EMS) has expanded from the urgent care of emergency patients to on-call healthcare services provided in the field with a holistic view of the patient’s wellbeing. This challenges EMS to find solutions to cover all demands, while simultaneously setting high skill requirements for EMS personnel. Understanding personnel is a critical element in developing functional and resistant EMS. The aim of this study was to investigate how Finnish EMS personnel emphasize the Emergency Medical Services Role Identity Scale aspects of caregiving, thrill-seeking, duty, and capacity; and if these role identities are associated with intention to leave the profession.MethodsWe conducted a cross-sectional survey (N = 616, 52% women, mean age 32.9 years). Data were collected through social media platforms and analyzed with means, standard deviations, Mann-Whitney U-tests, Kruskal-Wallis H-tests, and binary logistic regression analyses.ResultsOur results indicate that capacity is the most emphasized aspect among EMS personnel, and at the same time, it increases intention to leave EMS. Capacity was followed by caregiving, with no association with intention to leave. Duty and thrill-seeking were the least emphasized and were negatively associated with intention to leave. Additionally, there were also other factors that were associated with emphasizing EMS-RIS aspect and intention to leave.ConclusionCapacity stands out most strongly in analysis being at the core of the role identity of EMS personnel and was associated with a higher likelihood of leaving intentions. Several other factors were also associated with the intention to leave. Future studies should examine the exact dimensions of capacity that are considered important among EMS personnel and why factors such as work experience are associated with intentions to leave.

Correlation between Agglomerates Hausdorff Dimension and Mechanical Properties of Denture Poly(methyl methacrylate)-Based Composites

The microstructure–property relationship in poly(methyl methacrylate) PMMA composites is very important for understanding interface phenomena and the future prediction of properties that further help in designing improved materials. In this research, field emission scanning electron microscopy (FESEM) images of denture PMMA composites with SrTiO3, MnO2 and SrTiO3/MnO2 were used for fractal reconstructions of particle agglomerates in the polymer matrix. Fractal analysis represents a valuable mathematical tool for the characterization of the microstructure and finding correlation between microstructural features and mechanical properties. Utilizing the mathematical affine fractal regression model, the Fractal Real Finder software was employed to reconstruct agglomerate shapes and estimate the Hausdorff dimensions (HD). Controlled energy impact and tensile tests were used to evaluate the mechanical performance of PMMA-MnO2, PMMA-SrTiO3 and PMMA-SrTiO3/MnO2 composites. It was determined that PMMA-SrTiO3/MnO2 had the highest total absorbed energy value (Etot), corresponding to the lowest HD value of 1.03637 calculated for SrTiO3/MnO2 agglomerates. On the other hand, the highest HD value of 1.21521 was calculated for MnO2 agglomerates, while the PMMA-MnO2 showed the lowest Etot. The linear correlation between the total absorbed impact energy of composites and the HD of the corresponding agglomerates was determined, with an R2 value of 0.99486, showing the potential use of this approach in the optimization of composite materials’ microstructure–property relationship.

Many-body localization on finite generation fractal lattices

We study many-body localization in a hardcore boson model in the presence of random disorder on finite generation fractal lattices with different Hausdorff dimensions and different local lattice structures. In particular, we consider the Vicsek, T-shaped, Sierpinski gasket, and modified Koch-curve fractal lattices. In the single-particle case, these systems display Anderson localization for arbitrary disorder strength if they are large enough. In the many-body case, the systems available to exact diagonalization exhibit a transition between a delocalized and localized regime, visible in the spectral and entanglement properties of these systems. The position of this transition depends on the Hausdorff dimension of the given fractal, as well as on its local structure.

The Dimension of the Set of ψ-Badly Approximable Points in All Ambient Dimensions: On a Question of Beresnevich and Velani

Abstract Let $\psi :{\mathbb{N}} \to [0,\infty )$, $\psi (q)=q^{-(1+\tau )}$ and let $\psi $-badly approximable points be those vectors in ${\mathbb{R}}^{d}$ that are $\psi $-well approximable, but not $c\psi $-well approximable for arbitrarily small constants $c&amp;gt;0$. We establish that the $\psi $-badly approximable points have the Hausdorff dimension of the $\psi $-well approximable points, the dimension taking the value $(d+1)/(\tau +1)$ familiar from theorems of Besicovitch and Jarník. The method of proof is an entirely new take on the Mass Transference Principle (MTP) by Beresnevich and Velani (Annals, 2006); namely, we use the colloquially named “delayed pruning” to construct a sufficiently large $\liminf $ set and combine this with ideas inspired by the proof of the MTP to find a large $\limsup $ subset of the $\liminf $ set. Our results are a generalisation of some $1$-dimensional results due to Bugeaud and Moreira (Acta Arith, 2011), but our method of proof is nothing alike.

The influence of mindfulness meditation combined with progressive muscle relaxation training on the clinical efficacy and quality of life of patients with sarcopenia receiving haemodialysis: a randomised controlled trial

ObjectiveTo study the effect of mindfulness meditation combined with progressive muscle relaxation training on the clinical efficacy and quality of life in patients with sarcopenia receiving maintenance haemodialysis (MHD).MethodsEligible patients with sarcopenia in our hospital were randomly assigned to a control group (n = 24) and an intervention group (n = 25). The control group received conventional dialysis treatment, while the intervention group underwent mindfulness meditation combined with progressive muscle relaxation training during the interdialysis period in addition to conventional dialysis treatment. The effect of the intervention was evaluated after 12 weeks.ResultsThere were no significant differences in the baseline values of various parameters between the two groups. Exercise capacity (sit-to-stand test,handgrip,time to 10 sit-ups) significantly improved in the intervention group after 12 weeks (32.68 ± 8.32 vs 26.50 ± 6.83; 37.42 ± 10.12 vs 28.12 ± 8.51; 19.8 ± 5.40 vs 25.29 ± 7.18) (p < 0.05). In terms of the kidney disease quality of life (KDQOLTM) score, all other dimensions except sexual function, social functioning, burden of kidney disease and work status dimensions showed significant improvement compared to the baseline (p < 0.05). In the control group, only the dialysis staff encouragement (DSE) and patient satisfaction (PS) dimensions showed slight improvements compared to the baseline (p > 0.05). When compared with the control group, the intervention group showed significant improvements in 10 dimensions of exercise capacity and KDQOLTM scores for physical function, role-physical, general health, energy, symptom/problem list, sleep, DSE, pain, cognitive function, emotional well-being and patient PS after 12 weeks (61.30 ± 5.38 vs 42.98 ± 5.73; 57.50 ± 3.55 vs 50.70 ± 3.62) (p < 0.05). Some inflammatory markers, such as the levels of interleukin-6 and high-sensitivity C-reactive protein (30.29 ± 2.96 vs 17.65 ± 3.22; 8.93 ± 0.99 vs 3.02 ± 0.34), showed a decrease during the intervention, while albumin and prealbumin levels were significantly increased compared with the baseline (30.62 ± 1.65 vs 35.60 ± 1.68; 0.32 ± 0.05 vs 0.44 ± 0.07) (p < 0.05).ConclusionCombined intervention training can improve the motor ability and quality of life of patients with sarcopenia within a short period of time.

Finite dimension of the ion pathway networks in conducting glasses

In disordered materials, the ordinary understanding is that charge carriers tend to occupy energetically favorable sites known as ion-conducting channels. Many studies have revealed that the inherent fractal properties of such pathways lead to a sub-diffusive behavior. The linearity or branching of these pathways is crucial for determining how the charge carriers move. It can be thought that as the space dimensionality decreases, the average distance between the highest energy barriers along the conduction paths increases. In this study the finite dimension of those pathways is computed using an extended version of the classical Hausdorff dimension. Also, the Arrhenius behavior of the most mobile lithium ions is proved, confirming that such are responsible for conductivity behavior. The lithium ions mobility behavior in response to temperature changes and the finite dimension allowed to identify the ion diffusion regions fractal features. The reported results demonstrate that as the temperature increases the conducting channels become broadener, facilitating the transfer of electrical charge through the glassy matrix, below the transition temperature. The pathways behavior confirms the increase of the ionic conductivity when the temperature increases as it is experimentally observed.

Regularity of the Hausdorff dimension of some hyperbolic systems

We study the regularity of the Hausdorff dimension for certain hyperbolic basic sets, as a function of the dynamical system. The main results ensure that the Hausdorff dimension of certain multidimensional thin hyperbolic attractors and of certain nonconformal iterated function systems varies smoothly and analytically with respect to the diffeomorphism. Similar results hold for average conformal hyperbolic sets and average conformal repellers.

Regularity of the Optimal Sets for a Class of Integral Shape Functionals

We prove the first regularity theorem for the free boundary of solutions to shape optimization problems involving integral functionals, for which the energy of a domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is obtained as the integral of a cost function j(u, x) depending on the solution u of a certain PDE problem on Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}. The main feature of these functionals is that the minimality of a domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} cannot be translated into a variational problem for a single (real or vector valued) state function. In this paper we focus on the case of affine cost functions j(u,x)=-g(x)u+Q(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j(u,x)=-g(x)u+Q(x)$$\\end{document}, where u is the solution of the PDE -Δu=f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\Delta u=f$$\\end{document} with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal u from the inwards/outwards optimality of Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} and then we use the stability of Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} with respect to variations with smooth vector fields in order to study the blow-up limits of the state function u. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose ∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega $$\\end{document} into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of ∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega $$\\end{document} we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove C∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^\\infty $$\\end{document} regularity of the regular part of the free boundary when the data are smooth.

On a lower bound of Hausdorff dimension of weighted singular vectors

AbstractLet be a ‐tuple of positive real numbers such that and . A ‐dimensional vector is said to be ‐singular if for every , there exists such that for all , the system of inequalities has an integer solution . We prove that the Hausdorff dimension of the set of ‐singular vectors in is bounded below by . Our result partially extends the previous result of Liao et al. [Hausdorff dimension of weighted singular vectors in , J. Eur. Math. Soc. 22 (2020), 833–875].

Weighted topological pressure revisited

Abstract Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9)106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. &amp; Dynam. Sys.43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

Revisiting the IOM Reports and Envisioning a Promising Future for Public Health.

Two public health reports from the National Academies of Sciences, Engineering, and Medicine published in 1988 and 2003 by the Institute of Medicine continue to resonate. The COVID-19 pandemic highlighted the need for a robust and adequately funded public health system that has political and public support as well as strong connections to health care and other sectors. However, a spate of recent assessments of the nation's public health infrastructure shows continuing gaps in funding, workforce, capacity, and other dimensions. There are reasons for optimism and opportunities for progress in public health in the third decade of the 21st century. There is great promise in cross-sector partnerships and in embracing the "public" in public health by building power with communities in health improvement efforts and in decision-making. (Am J Public Health. 2024;114(5):495-500. https://doi.org/10.2105/AJPH.2024.307584).

Note on quasisymmetric rigidity of carpets

We prove in a constructive way that, given α>1, there exists a quasisymmetrically rigid metric carpet of Hausdorff dimension >α and whose peripheral circles are all rectifiable.

A Peak Set of Hausdorff Dimension 2n − 1 for the Algebra A(D) in the Boundary of a Domain D with C⌃2 Boundary

We consider a bounded strictly pseudoconvex domain Ω⊂Cn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {C}^{n}$$\\end{document} with C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{2}$$\\end{document} boundary. Then, we show that any compact Ahlfors–David regular subset of ∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega $$\\end{document} of Hausdorff dimension β∈(0,2n-1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\in (0,2n-1]$$\\end{document} contains a peak set E of Hausdorff dimension equal to β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}.