Related papers: Two-Scale Frostman Measures

In this paper, I will show how to use $\beta$-deformations to deal with dual flatness of $(\alpha,\beta)$-metrics. It is a natural continuation of the research on dually flat Randers metrics(see arxiv:1209.1150). $\beta$-deformations is a…

We investigate a quasisymmetrically invariant counterpart of the topological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension, gives a lower bound on the topological Hausdorff dimension of…

The notion of a Douglas space of second kind of a Finsler space with $(\alpha, \beta)$-metric was introduced by I. Y. Lee [9]. Since then, so many geometers have studied this topic e. g., [14]. In this paper, we prove that a Douglas space…

We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \subset {\Bbb R}^d$, $d \ge 2$, is greater than $\frac{d+1}{2}$, then the distance set $\Delta(E)=\{|x-y|: x,y \in E \}$ contains…

Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is,…

Given a compact set $E \subset \mathbb{R}^{d - 1}$, $d \geq 1$, write $K_{E} := [0,1] \times E \subset \mathbb{R}^{d}$. A theorem of C. Bishop and J. Tyson states that any set of the form $K_{E}$ is minimal for conformal dimension: if…

Let $B$ be a $d$-dimensional Gaussian process on $\mathbb{R}$, where the component are independents copies of a scalar Gaussian process $B_0$ on $\mathbb{R}_+$ with a given general variance function…

Several Riemannian metrics and families of Riemannian metrics were defined on the manifold of Symmetric Positive Definite (SPD) matrices. Firstly, we formalize a common general process to define families of metrics: the principle of…

Let $\mu$ be a Frostman measure on $E\subset\mathbb{R}^d$. The spherical average decay $$\int_{S^{d-1}}|\widehat{\mu}(r\omega)|^2\,d\omega\lesssim r^{-\beta} $$ was originally used to attack Falconer distance conjecture, via Mattila's…

Let $x=[a_1(x),a_2(x),\ldots]$ be the continued fraction expansion of $x\in[0,1)$. We prove that the Hausdorff dimension of \begin{equation*}E_{even}=\{x\in[0,1)\colon a_{2n}(x)\to\infty\ (n\to\infty)\}.\end{equation*} is 1/2. In general,…

We study sharpness of various generalizations of Frostman's lemma. These generalizations provide better estimates for the lower Hausdorff dimension of measures. As a corollary, we prove that if a generalized anisotropic gradient…

For a set $E \subseteq \mathbb{F}_q^d$, the distance set is defined as $\Delta(E) := \{\|\mathbf{x} - \mathbf{y}\| : \mathbf{x}, \mathbf{y} \in E\}$, where $\|\cdot\|$ denotes the standard quadratic form. We investigate the…

To any spectral triple (A,D,H) a dimension d is associated, in analogy with the Hausdorff dimension for metric spaces. Indeed d is the unique number, if any, such that |D|^-d has non trivial logarithmic Dixmier trace. Moreover, when d is…

We define a one-parameter family of canonical volume measures on Lorentzian (pre-)length spaces. In the Lorentzian setting, this allows us to define a geometric dimension - akin to the Hausdorff dimension for metric spaces - that…

In recent work, Harman and Snowden introduced a notion of measure on a Fra\"iss\'e class $\mathfrak{F}$, and showed how such measures lead to interesting tensor categories. Constructing and classifying measures is a difficult problem, and…

The fractal or Hausdorff dimension is a measure of roughness (or smoothness) for time series and spatial data. The graph of a smooth, differentiable surface indexed in $\mathbb{R}^d$ has topological and fractal dimension $d$. If the surface…

This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential theoretic methods are used to produce dimension bounds for images of sets under H\"older maps and…

We study Fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems. We prove that, under a mild technical condition, the Beurling dimension of a Fourier frame coincides with the Hausdorff…

Hove, Mo, and Sudbo [Phys. Rev. Lett. 85, 2368 (2000)] derived a simple connection, $\eta + D_H = 2$, between the anomalous scaling dimension $\eta$ of the U(1) universality class order parameter and the Hausdorff dimension $D_H$ of…

We extend classical results of Bridgeman-Taylor and McMullen on the Hessian of the Hausdorff dimension on quasi-Fuchsian space to the class of (1,1,2)-hyperconvex representations, a class introduced in arXiv:1902.01303 which includes small…