Related papers: Exceptional projections of sets exhibiting almost …
We consider several classical results related to the Hausdorff dimension of exceptional sets of orthogonal projections and try to find out whether they have reasonable formulations in terms of packing dimension. We also investigate the…
We present several applications of the Assouad dimension, and the related quasi-Assouad dimension and Assouad spectrum, to the box and packing dimensions of orthogonal projections of sets. For example, we show that if the (quasi-)Assouad…
Let $\gamma:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(\gamma(\theta),\gamma'(\theta),\gamma''(\theta)\big)\neq 0$. For each $\theta\in[0,1]$, let…
We generalize Bourgain's discretized projection theorem to higher rank situations. Like Bourgain's theorem, our result yields an estimate for the Hausdorff dimension of the exceptional sets in projection theorems formulated in terms of…
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…
In this paper, we study the quasisymmetric packing minimality of homogeneous perfect sets, and obtain that a special class of homogeneous perfect sets with $\operatorname{dim}_{P}E=1$ is quasisymmetrically packing minimal.
We extend the results previously published on exact packing dimensions of random recursive constructions to include constructions satisfying commonly occurring conditions. We remove the restrictive assumption that the diameter reduction…
In this article we prove the topological minimality of unions of several almost orthogonal planes of arbitrary dimensions. A particular case was proved in arXiv:1103.1468, where we proved the Almgren minimality (which is a weaker property…
Nearly orthogonal lattices were formally defined in [4], where their applications to image compression were also discussed. The idea of ``near orthogonality" in $2$-dimensions goes back to the work of Gauss. In this paper, we focus on…
Marstrand's celebrated projection theorem gives the Hausdorff dimension of the orthogonal projection of a Borel set in Euclidean space for almost all orthogonal projections. It is straightforward to see that sets for which the Fourier and…
We prove that, for every polyhedral or $C^1$ norm on $\mathbb{R}^d$ and every set $E \subseteq \mathbb{R}^d$ of packing dimension $s$, the packing dimension of the distance set of $E$ with respect to that norm is at least $\tfrac{s}{d}$.…
We study the packing dimension of Borel measures under orthogonal projections. We give a necessary and sufficient condition such that typical projections of Borel probability measures have full packing dimension and derive general lower…
We consider a problem posed by Erd\H{o}s, Herzog and Piranian on the maximum product of distances of a point set of order $n$ with a given diameter. We prove that it is sufficient to consider convex polygons and obtain results on the…
In this paper, we will discuss the notion of almost orthogonality in a functional sequence.Especially, we will define a few sequences of almost orthogonal polynomials which can be used successfully for modeling of electronic systems which…
We develop almost-orthogonality principles for maximal functions associated with averages over line segments and directional singular integrals. Using them, we obtain sharp $L^2$-bounds for these maximal functions when the underlying…
We prove a generalization of a result of Peres and Schlag on the dimensions of certain exceptional sets of projections and then apply it to a geometric problem.
We introduce an algorithm of joint approximation of a function and its first derivative by alternative orthogonal polynomials on the interval [0,1].The algorithm exhibits properties of shape preserving approximation for the function. A weak…
We study the projections in vector spaces over finite fields. We prove finite fields analogues of the bounds on the dimensions of the exceptional sets for Euclidean projection mapping. We provide examples which do not have exceptional…
The concept of \emph{almost orthogonal vectors}, i.e.\ vectors whose cosine similarity is close to $0$, relates to topics both in pure mathematics and in coding theory under the guises of spherical packing and spherical codes. In recent…
We extend Falconer's 1988 landmark result on the dimensions of self-affine fractals to encompass the dimensions of their projections, showing furthermore that their families of exceptional projections contain algebraic varieties which are…