Related papers: Projection Theorems for $\Phi$-Intermediate Dimens…

$\theta$ intermediate dimensions are a continuous family of dimensions that interpolate between Hausdorff and Box dimensions of fractal sets. In this paper we study the problem of the relationship between the dimension of a set…

Intermediate dimensions were recently introduced by Falconer, Fraser, and Kempton [Math. Z., 296, (2020)] to interpolate between the Hausdorff and box-counting dimensions. In this paper, we show that for every subset $ E $ of the symbolic…

Hausdorff $\Phi$-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce an effective version of Hausdorff $\Phi$-dimension, which we call constructive $\Phi$-dimension. We prove a…

Marstrand's projection theorem from $1954$ states that if $K \subset \mathbb{R}^{3}$ is an analytic set, then, for $\mathcal{H}^{2}$ almost every $e \in S^{2}$, the orthogonal projection $\pi_{e}(K)$ of $K$ to the line spanned by $e$ has…

We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if $\Lambda$ is such a carpet, and certain natural irrationality…

Marstrand's theorem states that applying a generic rotation to a planar set $A$ before projecting it orthogonally to the $x$-axis almost surely gives an image with the maximal possible dimension $\min(1, \dim A)$. We first prove, using the…

It is established a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower $Q$-homeomorphisms $f$ between domains in $\overline{\Rn}=\Rn\cup\{\infty\}$, $n\geqslant2$, under integral constraints…

We prove that for an arbitrary upper semi-continuous function $\phi\colon G(1,2) \to [0,1]$ there exists a compact set $F$ in the plane such that $\dim_{\textrm{A}} \pi F = \phi(\pi)$ for all $\pi \in G(1,2)$, where $\pi F$ is the…

Given a compact set $E\subset\mathbb{R}^d$ we investigate for which values of $m$ we have that $\dim_\theta P_V(E)=m$ or $\dim_\theta P_V(E)=\dim_\theta E$ for $\gamma_{d,m}-$almost all $V\in G(d,m)$. Our result can be extended to more…

The $\phi$-Assouad dimensions are a family of dimensions which interpolate between the upper box and Assouad dimensions. They are a generalization of the well-studied Assouad spectrum with a more general form of scale sensitivity that is…

This article surveys the $\theta$-intermediate dimensions that were introduced recently which provide a parameterised continuum of dimensions that run from Hausdorff dimension when $\theta=0$ to box-counting dimensions when $\theta=1$. We…

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.

In this paper, we introduce the mean $\Psi$-intermediate dimension which has a value between the mean Hausdorff dimension and the metric mean dimension, and prove the equivalent definition of the mean Hausdorff dimension and the metric mean…

We describe some recent results on the dimensions of linear projections of self-affine fractals, focusing in particular on an upper bound for the dimension of the projected image. We give a self-contained treatment of this bound and…

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 is concerned with restricted families of projections in $\mathbb{R}^{3}$. Let $K \subset \mathbb{R}^{3}$ be a Borel set with Hausdorff dimension $\dim K = s > 1$. If $\mathcal{G}$ is a smooth and sufficiently well-curved…

We present strong versions of Marstrand's projection theorems and other related theorems. For example, if E is a plane set of positive and finite s-dimensional Hausdorff measure, there is a set X of directions of Lebesgue measure 0, such…

This work is devoted to the study of families of infinite parabolic iterated function systems (PIFS) on a closed interval parametrized by vectors in $\mathbb{R}^d$ with overlaps. We show that the Hausdorff dimension and absolute continuity…

We study parametrized families of orthogonal projections for which the dimension of the parameter space is strictly less than that of the Grassmann manifold. We answer the natural question of how much the Hausdorff dimension may decrease by…

Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems…