Related papers: Falconer's problem for dot product on paraboloids

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…

We prove that on a $d$-dimensional Riemannian manifold, the distance set of a Borel set $E$ has a positive Lebesgue measure if $$\dim_{\mathcal{H}}(E)>\frac d2+\frac14+\frac{1-(-1)^d}{8d}.$$

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 study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dimension and spectrum. Interestingly, under reasonably mild…

A set in d dimensional Euclidean space with d larger than 2 having Hausdorff dimension at least d/2 must have distance set with Hausdorff dimension strictly greater than 1/2.

We present a family of sharpness examples for Falconer-type single dot product results. In particular, for $d\geq 2,$ for any $s<\frac{d+1}{2},$ we construct a Borel probability measure $\mu$ satisfying the energy estimate…

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 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…

The distinct dot products problem, a variant of the Erd\H{o}s distinct distances problem, asks "Given a set $P_n$ of $n$ points in $\mathbb{R}^2$, what is the minimum number $|D(P_n)|$ of distinct dot products they determine?" The best…

We are interested to bound from below the number of distinct dot products determined by a finite set of points $P$ in the Euclidean plane. In this paper, we build on the work of B. Hanson, O. Roche-Newton, and S. Senger, to obtain the…

Let $P \subset \mathbb R^2$ be a point set with cardinality $N$. We give an improved bound for the number of dot products determined by $P$, proving that, \[ |\{ p \cdot q :p,q \in P \}| \gg N^{2/3+c}. \] A crucial ingredient in the proof…

We study the complexity of geometric problems on spaces of low fractal dimension. It was recently shown by [Sidiropoulos & Sridhar, SoCG 2017] that several problems admit improved solutions when the input is a pointset in Euclidean space…

In this paper we study the following multi-parameter variant of the celebrated Falconer distance problem. Given ${\textbf{d}}=(d_1,d_2, \dots, d_{\ell})\in \mathbb{N}^{\ell}$ with $d_1+d_2+\dots+d_{\ell}=d$ and $E \subseteq \mathbb{R}^d$,…

We prove that if $E\subseteq \R^2$ is analytic and $1<d < \dim_H(E)$, there are ``many'' points $x\in E$ such that the Hausdorff dimension of the pinned distance set $\Delta_x E$ is at least $d\left(1 -…

Motivated by quite recent research involving the relationship between the dimension of a poset and graph-theoretic properties of its cover graph, we show that for every $d\geq 1$, if $P$ is a poset and the dimension of a subposet $B$ of $P$…

We prove that if the Hausdorff dimension of $E \subset {\Bbb R}^d$, $d \ge 3$, is greater than $\min \left\{ \frac{dk+1}{k+1}, \frac{d+k}{2} \right\},$ then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$, the set of…

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 prove that if the Hausdorff dimension of $E\subset\mathbb{R}^d$, $d\geq 2$ is greater than $\frac{d}{2}+\frac{1}{3}$, the set of gaps of $2$-chains inside $E$, $$\Delta_2(E)=\{(|x-y|, |y-z|): x, y, z\in E \}\subset\mathbb{R}^2$$ has…

We improve the Peres-Schlag result on pinned distances in sets of a given Hausdorff dimension. In particular, for Euclidean distances, with $$\Delta^y(E) = \{|x-y|:x\in E\},$$ we prove that for any $E, F\subset{\Bbb R}^d$, there exists a…

Given a finite point set $P$ in ${\mathbb R}^d$, and $\epsilon>0$ we say that $N\subseteq{ \mathbb R}^d$ is a weak $\epsilon$-net if it pierces every convex set $K$ with $|K\cap P|\geq \epsilon |P|$. We show that for any finite point set in…