Related papers: Intersections of middle-$\alpha$ Cantor sets with …

For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David-Semmes uniformization theorem states that if a compact metric space satisfies…

Let {\Omega} be a bounded domain in R^n with C^{1,1} boundary and let u_{\lambda} be a Neumann Laplace eigenfunction in {\Omega} with eigenvalue {\lambda}. We show that the (n - 1)-dimensional Hausdorff measure of the zero set of…

We consider (not self-similar) Cantor sets defined by a sequence of piecewise linear functions. We prove that the dimension of the harmonic measure on such a set is strictly smaller than its Hausdorff dimension. Some Hausdorff measure…

Let $ 1<\beta< 2 $, the sequence $\alpha(\beta)=\alpha(\beta)_1\alpha(\beta)_2\dotsb $ be the quasi-greedy $ \beta $-expansion of $ 1 $, and $ t\in [0,1) $ be a bifurcation parameter. The $\beta$-transformation is defined to be…

We show that in $\mathbb{R}^d$ there are purely unrectifiable sets of Hausdorff (and even box counting) dimension $d-1$ which are not tube null, settling a question of Carbery, Soria and Vargas, and improving a number of results by the same…

For any $0 < \alpha <1$, we construct Cantor sets on the parabola of Hausdorff dimension $\alpha$ such that they are Salem sets and each associated measure $\nu$ satisfies the estimate $\|{\widehat{f d\nu}}\|_{L^p(\mathbb{R}^2)} \leq C_p…

To each $\alpha\in(1/3,1/2)$ we associate the Cantor set $$\Gamma_{\alpha}:=\Big\{\sum_{i=1}^{\infty}\epsilon_{i}\alpha^i: \epsilon_i\in\{0,1\},\,i\geq 1\Big\}.$$ In this paper we consider the intersection $\Gamma_\alpha \cap (\Gamma_\alpha…

We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections.

We establish a formula yielding the Hausdorff measure for a class of non-self-similar Cantor sets in terms of the canonical covers of the Cantor set.

We study an interval exchange transformation of [0,1] formed by cutting the interval at the points 1/n and reversing the order of the intervals. We find that the transformation is periodic away from a Cantor set of Hausdorff dimension zero.…

For $2\leq p<\infty$, $\alpha'>2/p$, and $\delta>0$, we construct Cantor-type measures on $\mathbb{R}$ supported on sets of Hausdorff dimension $\alpha<\alpha'$ for which the associated maximal operator is bounded from $L^p_\delta…

We extend results by B. Hasselblatt, J. Schmeling in \emph{Dimension product structure of hyperbolic sets} (2004), and by the third author and K. Simon in \emph{Hausdorff and packing measures for solenoids} (2003), for $C^{1+\varepsilon}$…

We consider the Hausdorff dimension of planar Besicovitch sets for rectifiable sets $\Gamma$, i.e. sets that contain a rotated copy of $\Gamma$ in each direction. We show that for a large class of Cantor sets $C$ and Cantor-graphs $\Gamma$…

We introduce a one-parameter family of Borel regular measures on $\mathbb{R}^n$ that enhances Lebesgue measure by incorporating a scale-invariant penalty for codimension-1 boundary structures. Utilizing Carath\'eodory's outer measure…

We study the sets $\mathcal{L}$ and $\mathcal{M}\setminus\mathcal{L}$ near $3$, where $\mathcal{L}$ and $\mathcal{M}$ are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence…

Given $\ut\in\Rm$ and any norm $\Vert.\Vert$ on $\Rm$, we consider "inhomogeneously singular" vectors in $\Rm$ that admit an integer vector solution $(q,\underline{p})=(q,p_1,\ldots,p_m)$ to the system \[ 1\leq q\leq Q, \qquad \Vert…

A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set $K\subseteq \mathbb{R}$ is constructed such that every set definable in $(\mathbb{R},<,+,\cdot,K)$ is Borel. In addition, we…

In this paper we consider the specification property for $(\alpha,\beta)$-shifts. When $\alpha=0$, Schmeling shows that the set of $\beta>1$ for which the $\beta$-shift has the specification property has the Lebesgue measure zero but has…

We prove that the dimension of the harmonic measure of the complementary of a translation-invariant type of Cantor sets as a continuous function of the parameters determining these sets. This results extend a previous one of the author and…

We consider the question which compact metric spaces can be obtained as a Lipschitz image of the middle third Cantor set, or more generally, as a Lipschitz image of a subset of a given compact metric space. In the general case we prove that…