Intermediate value property for the Assouad dimension of measures

Abstract Hare, Mendivil, and Zuberman have recently shown that if $X \subset \mathbb{R}$ is compact and of non-zero Assouad dimension $\dim_{A} X$, then for all $s > \dim_{A} X$, $X$ supports measures with Assouad dimension $s$. We generalize this result to arbitrary complete metric spaces

Fractal percolation and quasisymmetric mappings

Abstract We study the conformal dimension of fractal percolation and show that, almost surely, the conformal dimension of a fractal percolation is strictly smaller than its Hausdorff dimension

Projections of Poisson cut-outs in the Heisenberg group and the visual 3-sphere

Abstract We study projectional properties of Poisson cut-out sets $E$ in non-Euclidean spaces. In the first Heisenbeg group $\mathbb{H} =\mathbb{C}×\mathbb{R}$, endowed with the Korányi metric, we show that the Hausdorff dimension of the vertical projection $π(E)$ (projection along the center of $\mathbb{H}$) almost surely equals $\textrm{min}\{2, \textrm{dim}_{\textrm{H}}(E)\}$ and that $π(E)$ has non-empty interior if $\textrm{dim}_{\textrm{H}}(E) > 2$. As a corollary, this allows us to determine the Hausdorff dimension of $E$ with respect to the Euclidean metric in terms of its Heisenberg Hausdorff dimension $\textrm{dim}_{\textrm{H}}(E)$. We also study projections in the one-point compactification of the Heisenberg group, that is, the 3-sphere $\mathbf{S}^3$ endowed with the visual metric $d$ obtained by identifying $\mathbf{S}^3$ with the boundary of the complex hyperbolic plane. In $\mathbf{S}^3$, we prove a projection result that holds simultaneously for all radial projections (projections along so called “chains”). This shows that the Poisson cut-outs in $\mathbf{S}^3$ satisfy a strong version of the Marstrand’s projection theorem, without any exceptional directions

New bounds on Cantor maximal operators

Abstract We prove $L^p$ bounds for the maximal operators associated to an Ahlfors-regular variant of fractal percolation. Our bounds improve upon those obtained by I. Łaba and M. Pramanik and in some cases are sharp up to the endpoint. A consequence of our main result is that there exist Ahlfors-regular Salem Cantor sets of any dimension > 1/2 such that the associated maximal operator is bounded on $L^2(\mathbb{R})$. We follow the overall scheme of Łaba–Pramanik for the analytic part of the argument, while the probabilistic part is instead inspired by our earlier work on intersection properties of random measures

Patterns in random fractals

Abstract We characterize the existence of certain geometric configurations in the fractal percolation limit set A in terms of the almost sure dimension of A. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemerédi theorems for random discrete sets, we also consider the corresponding problem for sets of positive ν-measure, where ν is the natural measure on A. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating the intersections of the products of m independent realizations of A with transversal planes and, more generally, algebraic varieties, and extend some well-known features of independent percolation on trees to a setting with long-range dependencies

Spatially independent martingales, intersections, and applications

Abstract We define a class of random measures, spatially independent martingales, which we view as a natural generalization of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrized measures {ηt}t, and show that under some natural checkable conditions, a.s. the mass of the intersections is Hölder continuous as a function of t. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. Laba in connection to the restriction problem for fractal measures

Random cutout sets with spatially inhomogeneous intensities

Abstract We study the Hausdorff dimension of Poissonian cutout sets defined via inhomogeneous intensity measures on Q-regular metric spaces. Our main results explain the dependence of the dimension of the cutout sets on the multifractal structure of the average densities of the Q-regular measure. As a corollary, we obtain formulas for the Hausdorff dimension of such cutout sets in self-similar and self-conformal spaces using the multifractal decomposition of the average densities for the natural measures

Fractal percolation, porosity, and dimension

Abstract We study the porosity properties of fractal percolation sets E ⊂ Rd. Among other things, for all 0 < ɛ < ½, we obtain dimension bounds for the set of exceptional points where the upper porosity of E is less than ½ ‒ ɛ, or the lower porosity is larger than ɛ. Our method works also for inhomogeneous fractal percolation and more general random sets whose offspring distribution gives rise to a Galton–Watson process

Hausdorff dimension of limsup sets of random rectangles in products of regular spaces

Abstract The almost sure Hausdorff dimension of the limsup set of randomly distributed rectangles in a product of Ahlfors regular metric spaces is computed in terms of the singular value function of the rectangles

On dimensions of visible parts of self-similar sets with finite rotation groups

Abstract We derive an upper bound for the Assouad dimension of visible parts of self-similar sets generated by iterated function systems with finite rotation groups and satisfying the weak separation condition. The bound is valid for all visible parts and it depends on the direction and the penetrable part of the set, which is a concept defined in this paper. As a corollary, we obtain in the planar case that if the projection is a finite or countable union of intervals then the visible part is 1-dimensional. We also prove that the Assouad dimension of a visible part is strictly smaller than the Hausdorff dimension of the set provided the projection contains interior points. Our proof relies on Furstenberg’s dimension conservation principle for self-similar sets