Related papers: On the Pinned Distances Problem in Positive Charac…

We adapt the approach of Rudnev, Shakan, and Shkredov to prove that in an arbitrary field $\mathbb{F}$, for all $A \subset \mathbb{F}$ finite with $|A| < p^{1/4}$ if $p:= Char(\mathbb{F})$ is positive, we have $$|A(A+1)| \gtrsim |A|^{11/9},…

Suppose $E, F$ are Borel sets in the plane, $\dim_{\mathcal{H}} E>1$, $\dim_{\mathcal{H}} E+\dim_{\mathcal{H}} F>2$, and $F$ has equal Hausdorff and packing dimension. We prove that there exists $y\in F$ such that the pinned distance set…

In this paper, we give improved bounds on the Hausdorff dimension of pinned distance sets of planar sets with dimension strictly less than one. As the planar set becomes more regular (i.e., the Hausdorff and packing dimension become…

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…

We prove that a finite subset $A$ of the projective line $FP^1$ over a field $F$ of positive characteristic $p$ determines $\Omega(|A|^2)$ distinct cross-ratios, as long as $|A|<\sqrt{p}.$

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…

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 study a variant of the Erd\H os unit distance problem, concerning angles between successive triples of points chosen from a large finite point set. Specifically, given a large finite set of $n$ points $E$, and a sequence of angles…

In this paper, we prove Erd\H{o}s distance conjecture in $\mathbb{R}^d$, namely, a set of $n$ points in $\mathbb{R}^2$ determines $\Omega(\frac{n}{\sqrt{\log n}})$ distances, and for $d\ge 3$, a set of $n$ points in $\mathbb{R}^d$…

We study a finite-field analogue of the Erd\H{o}s distinct distances problem under the Hamming metric. For a set \(S\subseteq \mathbb{F}_q^n\) let $\Delta(S)$ denote the set of Hamming distances determined by \(S\). We prove the lower bound…

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…

Federer's characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension one. In complete…

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…

Let $\mathbb{F}_p$ be a prime field, and ${\mathcal E}$ a set in $\mathbb{F}_p^2$. Let $\Delta({\mathcal E})=\{||x-y||: x,y \in {\mathcal E} \}$, the distance set of ${\mathcal E}$. In this paper, we provide a quantitative connection…

We prove, under suitable conditions, a lower bound on the number of pinned distances determined by small subsets of two-dimensional vector spaces over fields. For finite subsets of the Euclidean plane we prove an upper bound for their…

We prove the following variant of the Falconer conjecture in the plane. If the dimension of a compact planar set is greater than one, then the distance set with respect to almost every ellipse has positive Lebesgue measure.

We investigate the computational complexity of computing the Hausdorff distance. Specifically, we show that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for…

In this paper, we study the distance problem in the setting of finite p-adic rings. In odd dimensions, our results are essentially sharp. In even dimensions, we clarify the conjecture and provide examples to support it. Surprisingly,…

Let $V\subset \mathbb{F}_q^d$ be a \textit{regular} variety, $k\ge 3$ is an integer and $A\subseteq V$. Covert, Koh, and Pi (2017) proved the following generalization of the Erd\H{o}s-Falconer distance problem: If $|A|\gg…