Let $d\geq 2$, $0<\alpha<1$, and let $\Gamma_k$ be the successive arrival times of a standard Poisson process on $(0,\infty)$. Given independent uniform directions $\varepsilon_k\in S^{d-1}$, independent of $(\Gamma_k)$, we consider the random countable stable zonotope $Z_\alpha=\bigoplus_{k=1}^{\infty}\Gamma_k^{-1/\alpha}[0,\varepsilon_k]$. For its set of extreme points $\operatorname{ext} Z_\alpha$, we prove that almost surely $\dim_H \operatorname{ext} Z_\alpha=(d-1)\alpha$, and that the critical Hausdorff measure $\mathcal H^{(d-1)\alpha}(\operatorname{ext} Z_\alpha)$ is almost surely finite. The lower bound follows from the tangential non-degeneracy of the stable increments of the parametrizing field and Frostman's energy criterion. For the upper bound we construct an adaptive covering: at each scale the Poisson jumps are split into large and small ones, the large jumps determine a finite hyperplane arrangement, and the sum of the small jumps controls the diameters of the images of its cells.
For $0<d\leq1/4$, let $\mathcal{C}(d)$ be the four-corner Cantor set with Hausdorff dimension $s_d=\log 4/\log(1/d)$. Peres, Simon, and Solomyak, as well as Mattila, asked for which $d$ the measure $\Hau^{s_d}(p_\theta(\mathcal{C}(d)))$ is positive for almost every direction $\theta$. It was open for the range $1/9\leq...
Let $\Lambda=\operatorname{diag}(m_1,\ldots,m_d)$ be a diagonal expanding integer matrix with $m_1\ge\cdots\ge m_d\ge2$ and at least two distinct diagonal entries, and let $X\subset\mathcal A^{\mathbb N}$ be a subshift with weak specification over $\mathcal A=\prod_{r=1}^d\{0,1,\ldots,m_r-1\}$. Let $K=R(X)$ be the corr...
Let $\Omega\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,\alpha}$, where $0<\alpha<1$. For the adjoint Neumann--Poincar\'e operator $K^*_{\partial\Omega}$, normalised so that its distinguished eigenvalue is $1/2$, let $\lambda_j^+(\Omega)$ denote the upper min--max values on...
Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In parti...
We introduce an explicit deterministic collection of $N$ spherical points, $\cP_N\subset\mathbb S^2$, that we call the {\em deterministic Diamond points}. For every $0<\alpha<2$ we prove that there exists a constant $C_\alpha>0$ such that \[ 0\leq \frac{2^{\alpha+1}}{\alpha+2}N^2-\sum_{x,y\in\cP_N}|x-y|^\alpha \le C_\a...
Carlos Beltrán, J. Marzo, J. Ortega-Cerdà· 0 citations
Let $X\subset \mathbb C^n$ be a closed pure $d$-dimensional complex analytic set. We associate to $X$ its set of projective asymptotic directions $$ \Sigma_\infty(X) :=\{\ell \in \mathbb P^{n-1}(\mathbb C):\ell\cap C_\infty(X)\ne\{0\}\}, $$ where $ C_\infty(X)$ is the total tangent cone at infinity. We prove the metric...
Nhan D. V. Nguyen· 1 citation· ⚡1
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