相关论文: Hausdorff dimension estimates for restricted famil…

This paper investigates a refinement of Marstrand's projection theorem; more specifically, let $\Pi_t, t\in[0,1]$ be a family of $m$ dimensional subspaces of the Euclidean space $\mathbb{R}^n$ and let $P_t:\mathbb{R}^4\mapsto \Pi_t$ be the…

We study the restricted families of orthogonal projections in $\mathbb{R}^3$. We show that there are families of random subspaces which admit a Marstrand- Mattila type projection theorem.

Let $\gamma: I \to S^2$ be a $C^2$ curve with $\det(\gamma, \gamma', \gamma'')$ nonvanishing, and for each $\theta \in I$ let $\rho_{\theta}$ be orthogonal projection onto the span of $\gamma(\theta)$. It is shown that if $A \subseteq…

This paper extends some results of [M5] and [M3], in particular, removing assumptions of positive lower density. We give conditions on a general family $P_{\lambda}:\mathbb{R}^{n}\to\mathbb{R}^{m}, \lambda \in \Lambda,$ of orthogonal…

We study dimensions of sets projected to an $(n-2)$-dimensional family of hyperplanes in $\mathbb{R}^n$ under curvature conditions. Let $n\ge 3$ and $\Sigma \subset S^{n-1}$ be an $(n-2)$-dimensional $C^2$ manifold such that $\Sigma$ has…

We show that, almost surely, the Hausdorff dimension $s_0$ of a random covering set is preserved under all orthogonal projections to linear subspaces with dimension $k>s_0$. The result holds for random covering sets with a generating…

We investigate how the Hausdorff dimension and measure of a self-similar set $K\subseteq\mathbb{R}^{d}$ behave under linear images. This depends on the nature of the group $\mathcal{T}$ generated by the orthogonal parts of the defining maps…

We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if $A\subset\mathbb{R}^2$ is a Borel set of Hausdorff dimension $s>1$, then its distance set has Hausdorff…

In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends…

We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than $1$, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In…

In the present work, we investigate existence of deformations and algebraic approximability for certain uniruled K\"ahler threefolds. In the first part, we establish existence of infinitesimal deformations for all conic bundles with…

We study projectional properties of Poisson cut-out sets $E$ in non-Euclidean spaces. In the first Heisenbeg group, endowed with the Kor\'anyi metric, we show that the Hausdorff dimension of the vertical projection $\pi(E)$ (projection…

Let {\mathbb{V} = V x R^l : V \in G(n-l,m-l)} be the family of m-dimensional subspaces of R^n containing {0} x R^l, and let \pi_{\mathbb{V}} : R^n --> \mathbb{V} be the orthogonal projection onto \mathbb{V}. We prove that the mapping V…

This is a survey on recent developments on the Hausdorff dimension of projections and intersections for general subsets of Euclidean spaces, with an emphasis on estimates of the Hausdorff dimension of exceptional sets and on restricted…

We establish several optimal estimates for exceptional parameters in the projection of fractal measures: (1) For a parametric family of self-similar measures satisfying a transversality condition, the set of parameters leading to a…

This paper contains new results on two classical topics in fractal geometry: projections, and intersections with affine planes. To keep the notation of the abstract simple, we restrict the discussion to the planar cases of our theorems. Our…

Intermediate dimensions were recently introduced to interpolate between the Hausdorff and box-counting dimensions of fractals. Firstly, we show that these intermediate dimensions may be defined in terms of capacities with respect to certain…

In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic manifold, we…

For a compact subset K of the plane and a point x, we define the visible part of K from x to be the set K_x={u\in K : [x,u]\cap K={u}}. (Here [x,u] denotes the closed line segment joining x to u.) In this paper, we use energies to show that…

We study the family of vertical projections whose fibers are right cosets of horizontal planes in the Heisenberg group, $\mathbb{H}^n$. We prove lower bounds for Hausdorff dimension distortion of sets under these mappings, with respect to…