Related papers: Density spectrum of Cantor measure

In this paper, we find a full Lebesgue measure set of frequencies $\check \II\subset [0,1]\setminus \Q$ such that for any $(\alpha,\lambda)\in \check \II\times [24,\infty)$, the Hausdorff and box dimensions of the spectrum of the Sturmian…

We initiate a study of the following problem: Given a continuous domain $\Omega$ along with its convex hull $\mathcal{K}$, a point $A \in \mathcal{K}$ and a prior measure $\mu$ on $\Omega$, find the probability density over $\Omega$ whose…

Given a weight vector $\tau=(\tau_{1}, \dots, \tau_{n}) \in \mathbb{R}^{n}_{+}$ with each $\tau_{i}$ bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set of $\tau$-approximable points points over a…

Consider a Moran-type iterated function system (IFS) \( \{\phi_{k,d}\}_{d\in D_{2p_k}, k\geq 1} \), where each contraction map is defined as \[ \phi_{k,d}(x) = (-1)^d b_k^{-1}(x + d), \] with integer sequences \( \{b_k\}_{k=1}^\infty \) and…

Let $\mu$ and $\nu$ be two Borel probability measures on two separable metric spaces $\X$ and $\Y$ respectively. For $h, g$ be two Hausdorff functions and $q\in \R$, we introduce and investigate the generalized pseudo-packing measure…

We present constraints on the deuterium to hydrogen ratio (D/H) in the metal-poor gas cloud at redshift $z=0.701$ towards QSO 1718+4807. We use new Keck spectra in addition to Hubble Space Telescope (HST) and International Ultraviolet…

We consider the case of hyperbolic basic sets $\Lambda$ of saddle type for holomorphic maps $f: \mathbb P^2\mathbb C \to \mathbb P^2\mathbb C$. We study equilibrium measures $\mu_\phi$ associated to a class of H\"older potentials $\phi$ on…

A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers, when represented using continued fractions, exhibit partial quotients that grow at specific rates. For any positive function…

We compute the exact Hausdorff and Packing measures of linear Cantor sets which might not be self similar or homogeneous . The calculation is based on the local behavior of the natural probability measure supported on the sets.

We study the spectral properties of discrete Schr\"odinger operators with potentials given by primitive invertible substitution sequences (or by Sturmian sequences whose rotation angle has an eventually periodic continued fraction…

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\Bbb{R}^n$, $n\geq 2$. To each locally finite Borel measure $\mu$, we associate a function $\widetilde J_2(\mu, x)$…

We give conditions on a general family $P_{\lambda}:\R^n\to\R^m, \lambda \in \Lambda,$ of orthogonal projections which guarantee that the Hausdorff dimension formula $\dim A\cap P_{\lambda}^{-1}\{u\}=s-m$ holds generically for measurable…

Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega: \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega>0$ is small. Assume also that $\Omega$ is twice…

This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let $u$ be a real-valued harmonic function in $\mathbb{R}^n$ with $u(0)=0$ and $n\geq 3$. We prove…

In this note we present an algorithm to obtain a uniform lower bound on Hausdorff dimension of the stationary measure of an affine iterated function scheme with similarities, the best known example of which is Bernoulli convolution. The…

We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These entropies…

In a recent paper, Baker and Kong have studied the Hausdorff dimension of the intersection of Cantor sets with their translations. We extend their results to more general Cantor sets. The proofs rely on the frequencies of digits in unique…

Let $C_1$ and $C_2$ be two Cantor sets with convex hull $[0,1]$. Newhouse proved if $\tau(C_1)\cdot \tau(C_2)\geq 1$, then the arithmetic sum $C_1+C_2$ is an interval, where $\tau(C_i), 1\leq i\leq 2$ denotes the thickness of $C_i$. In this…

We show that if a separable space X has a meager open subset containing a copy of the Cantor set 2^\omega, then X has $\frak{c}$ types of countable dense subsets. We suggest a generalization of the \lambda-set for non-separable spaces. Let…

It is a longstanding conjecture that given a subset $E$ of a metric space, if $E$ has finite Hausdorff measure in dimension $\alpha\ge 0$ and $\mathscr{H}^\alpha\llcorner E$ has unit density almost everywhere, then $E$ is an…