相关论文: Endpoint bounds for the non-isotropic Falconer dis…

The Falconer conjecture asserts that if E is a planar set with Hausdorff dimension strictly greater than 1, then its Euclidean distance set has positive one-dimensional Lebesgue measure. We discuss the analogous question with the Euclidean…

We show that if compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}$, where $d\geq 4$ is an even integer, then the distance set of $E$ has positive Lebesgue measure. This improves the previously…

The Falconer distinct distance problem asks for a compact set $E\subset\mathbb{R}^d$ how large its Hausdorff dimension needs to be to ensure that the Lebesgue measure of its distance set is positive. In this paper we consider the analogous…

The recent breakthrough of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set $A\subset \mathbb{R}^2$, if the Hausdorff dimension of $A$ is greater than $\frac{5}{4}$, then the distance set…

Falconer proved that there are sets $E\subset \mathbb{R}^n$ of Hausdorff dimension $n/2$ whose distance sets $\{|x-y| : x,y\in E\}$ are null with respect to Lebesgue measure. This led to the conjecture that distance sets have positive…

We show that if a compact set $E\subset \mathbb{R}^d$ has Hausdorff dimension larger than $\frac{d}{2}+\frac{1}{4}-\frac{1}{8d+4}$, where $d\geq 3$, then there is a point $x\in E$ such that the pinned distance set $\Delta_x(E)$ has positive…

We study the distance set problem for pairs of compact sets $A, B\subset \mathbb{R}^n$, $n\geq 2$. We show that if $B$ is contained in a hyperplane and \begin{align*} \dim_{H} A+\dim_{H} B>n, \end{align*} then the distance set $…

In this paper we study the following variant of the Falconer distance problem. Let $E$ be a compact subset of ${\mathbb{R}}^d$, $d \ge 1$, and define $$ \Box(E)=\left\{\sqrt{{|x-y|}^2+{|x-z|}^2}: x,y,z \in E,\, y\neq z \right\}.$$ We shall…

If $E \subset \mathbb{R}^2$ is a compact set of Hausdorff dimension greater than $5/4$, we prove that there is a point $x \in E$ so that the set of distances $\{ |x-y| \}_{y \in E}$ has positive Lebesgue measure.

A celebrated result due to Wolff says if $E$ is a compact subset of ${\Bbb R}^2$, then the Lebesgue measure of the distance set $\Delta(E)=\{|x-y|: x,y \in E \}$ is positive if the Hausdorff dimension of $E$ is greater than $\frac{4}{3}$.…

In this paper we investigate the Erd\"os/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been…

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $\alpha>0$ such that $|\Delta(E)| \gtrsim q$ whenever $|E| \gtrsim q^{\alpha}$, where $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector…

Let $1 \leq k \leq d$ and consider a subset $E\subset \mathbb{R}^d$. In this paper, we study the problem of how large the Hausdorff dimension of $E$ must be in order for the set of distinct noncongruent $k$-simplices in $E$ (that is,…

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 $E\subset \mathbb{R}^{d}$…

We establish dimensional thresholds for dot product sets associated with compact subsets of translated paraboloids. Specifically, we prove that when the dimension of such a subset exceeds $ \frac{5}{4} = \frac{3}{2} - \frac{1}{4} $ in…

In this paper, we study the cardinality of the distance set $\Delta(A, B)$ determined by two subsets $A$ and $B$ of the $d$-dimensional vector space over a finite field $\mathbb{F}_q$. Assuming that $A$ or $B$ lies in a $k$-coordinate plane…

We introduce a class of Falconer distance problems, which we call of restricted type, lying between the classical version and its pinned variant. Prototypical restricted distance sets are the diagonal distance sets, $k$-point configuration…

We prove some weighted refined decoupling estimates. As an application, we give an alternative proof of the following result on Falconer's distance set problem by the authors in a companion work: if a compact set $E\subset \mathbb{R}^d$ has…

In this paper we prove, for all $d \ge 2$, that for no $s<\frac{d+1}{2}$ does $I_s(\mu)<\infty$ imply the canonical Falconer distance problem incidence bound, or the analogous estimate where the Euclidean norm is replaced by the norm…

In this article, we study two problems concerning the size of the set of finite point configurations generated by a compact set $E\subset \mathbb{R}^d$. The first problem concerns how the Lebesgue measure or the Hausdorff dimension of the…