相关论文: Yet another proof of Marstrand's Theorem

In this paper we study multi-parameter projection theorems for fractal sets. With the help of these estimates, we recover results about the size of $A \cdot A+...+A \cdot A$, where $A$ is a subset of the real line of a given Hausdorff…

We show that for any compact set $E\subset\mathbb{R}^d$ the visible part of $E$ has Hausdorff dimension at most $d-1/6$ for almost every direction. This improves recent estimates of Orponen and Matheus. If $E$ is $s$-Ahlfors regular, where…

Under mild assumptions, we characterise modules with projective resolutions of length n in the target category of filtrated K-theory over a finite topological space in terms of two conditions involving certain Tor-groups. We show that the…

Firstly we show a generalization of the (1,1)-Lefschetz theorem for projective toric orbifolds and secondly we prove that on 2k-dimensional quasi-smooth hypersurfaces coming from quasi-smooth intersection surfaces, under the Cayley trick,…

We study an extension of the Falconer distance problem in the multiparameter setting. Given $\ell\geq 1$ and $\mathbb{R}^{d}=\mathbb{R}^{d_1}\times\cdots \times\mathbb{R}^{d_\ell}$, $d_i\geq 2$. For any compact set $E\subset \mathbb{R}^{d}$…

In this paper we establish a Besicovitch-Federer type projection theorem for general measures. Specifically, let $\mu$ be a finite Borel measure on $\mathbb{R}^n$ and let $0 < m < n$ be an integer. We show that, under the sole assumption…

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…

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…

Tarski's Circle Squaring Problem from 1925 asks whether it is possible to partition a disk in the plane into finitely many pieces and reassemble them via isometries to yield a partition of a square of the same area. It was finally resolved…

We prove a Kaufman-type exceptional set estimate for sets in $\mathbb{R}^n$ that have optimal oracles, a class of sets that strictly contains the analytic sets and sets with equal Hausdorff and packing dimension. As a consequence, we…

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…

Letting A be a Borel subset of n dimensional Euclidean space, and W(x) be an m dimensional affine subspace containing x and varying in a Lipschitz way according to x, we establish that A is Lebesgue null if and only if $A \cap W(x)$ has m…

We show that whenever a separable subset $S$ of a complete metric space $X$ admits a $d$-dimensional weak tangent field, the set $S$ is close to being $d$-dimensional in the following sense. Whenever $\mu$ is a Borel finite measure on $X$…

The Brunn-Minkowski Theory has seen several generalizations over the past century. Many of the core ideas have been generalized to measures. With the goal of framing these generalizations as a weighted Brunn-Minkowski theory, we prove the…

Let R be a commutative domain. We prove that an R-module B is projective if and only if Ext^1(B,T)=0 for any torsion module T. This answers in the affirmative a question raised by Kaplansky in 1962.

This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of $\mathbb{R}^{2n}$, as well as dimension of intersections of sets with isotropic planes. It is shown that if $A$ and $B$ are Borel subsets…

We prove that if the Hausdorff dimension of a compact subset of ${\mathbb R}^d$ is greater than $\frac{d+1}{2}$, then the set of angles determined by triples of points from this set has positive Lebesgue measure. Sobolev bounds for…

In this paper, we study the multifractal Hausdorff and packing dimensions of Borel probability measures and study their behaviors under orthogonal projections. In particular, we try through these results to improve the main result of M. Dai…

We prove a version of the Erd\H{o}s--Beck Theorem from discrete geometry for fractal sets in all dimensions. More precisely, let $X\subset \mathbb{R}^n$ Borel and $k \in [0, n-1]$ be an integer. Let $\dim (X \setminus H) = \dim X$ for every…

We prove a conjecture of H\'era on the dimension of unions of $k$-planes. Let $0<k \le d<n$ be integers, and $\beta\in[0,k+1)$. If $\mathcal{V}\subset A(k,n)$, with $\text{dim}(\mathcal{V})=(k+1)(d-k)+\beta$, then…