Le Anh Vinh
art, Iosevich, Koh and Rudnev (2007) show, using Fourier analysis method, that the finite Erdös-Falconer distance conjecture holds for subsets of the unit sphere in $\mathbbm{F}_q^d$. In this note, we give a graph theoretic proof of this result.
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A matrix variant of the Erdős-Falconer distance problems over finite field
本文研究有限域向量空间中Erdős-Falconer距离问题的矩阵类似物，并由此引发对某些二次矩阵Gauss和的有趣分析。
There exists an infinite family of examples of subsets of $\mathbb{F}_q^2$ with $q^{4/3}$ elements whose distance sets are not the whole of $\mathbb{F}_q$.
We consider a finite fields version of the Erdős-Falconer distance problem for two different sets. In a certain range for the sizes of the two sets we obtain results of the conjectured order of magnitude.
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 $Δ(A)$ has positive Lebesgue measure. In a very recent paper, Murphy, Petridis, Pham, Rudnev, and Stevens (2022) proved the prime field version of this result, namely, for $E\subset\mathbb{F}_p^2$ with $|E|\gg p^{5/4}$, there exist many points $x\in E$ such that the number of distinct distances from $x$ is at least $cp$. The main purpose of this paper is to pr…
We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+... +A \cdot A$, where A is a subset ${\mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdo…
Falconer lattice sets and the Erdos similarity problem
研究发现极薄集合满足Erdős相似性猜想，这些集合具有小对数维数，不包含缓慢衰减序列的仿射副本，但包含快速衰减序列，基于Falconer格点集和Bourgain定理提出新结果。
On Erdos-Falconer distance problem in even dimensions
研究偶数维的埃尔德什 - 法尔科纳距离问题，通过证明提取定理，将偶数维猜想归结为平面情形，在固定距离问题和三角形分布上得到改进阈值，在素域和任意有限域分别取得新记录。
We study the following two-parameter variant of the Erd\H os-Falconer distance problem. Given $E,F \subset {\Bbb F}_q^{k+l}$, $l \geq k \ge 2$, the $k+l$-dimensional vector space over the finite field with $q$ elements, let $B_{k,l}(E,F)$ be given by $$\{(\Vert x'-y'\Vert, \Vert x"-y" \Vert): x=(x',x") \in E, y=(y',y") \in F; x',y' \in {\Bbb F}_q^k, x",y" \in {\Bbb F}_q^l \}.$$ We prove that if $|E||F| \geq C q^{k+2l+1}$, then $B_{k,l}(E,F)={\Bbb F}_q \times {\Bbb F}_q$. Furthermore this result is sharp if $k$ is odd. For the case of $l=k=2$ and $q$ a prime with $q \equiv 3 \mod 4$ we get that…
We obtain nontrivial exponents for Erd\H os-Falconer type problems. Let $T_k(E)$ denote the set of distinct congruent $k$-dimensional simplexes determined by $(k+1)$-tuples of points from $E$. We prove that there exists $s_0(d)<d$ such that, if $E \subset {\Bbb R}^d,\, d \ge 2$, with $dim_{\mathcal H}(E)>s_0(d)$, then the ${k+1 \choose 2}$-dimensional Lebesgue measure of $T_k(E)$ is positive. Results were previously obtained for triangles in the plane \cite{GI12} and in higher dimensions \cite{GGIP12}. In this paper, we improve upon those exponents, using a group-theoretic method that sheds ne…
The Erd\H os-Falconer distance problem in $\mathbb{Z}_q^d$ asks one to show that if $E \subset \mathbb{Z}_q^d$ is of sufficiently large cardinality, then $Δ(E) := \{(x_1 - y_1)^2 + \dots + (x_d - y_d)^2 : x, y \in E\}$ satisfies $Δ(E) = \mathbb{Z}_q$. Here, $\mathbb{Z}_q$ is the set of integers modulo $q$, and $\mathbb{Z}_q^d$ is the free module of rank $d$ over $\mathbb{Z}_q$. We extend known results in two directions. Previous results were known only in the setting $q = p^{\ell}$, where $p$ is an odd prime, and as such only showed that all units were obtained in the distance set. We remove t…
Given $E \subseteq \mathbb{F}_q^d \times \mathbb{F}_q^d$, with the finite field $\mathbb{F}_q$ of order $q$ and the integer $d \ge 2$, we define the two-parameter distance set as $Δ_{d, d}(E)=\left\{\left(\|x_1-y_1\|, \|x_2-y_2\|\right) : (x_1,x_2), (y_1,y_2) \in E \right\}$. Birklbauer and Iosevich (2017) proved that if $|E| \gg q^{\frac{3d+1}{2}}$, then $ |Δ_{d, d}(E)| = q^2$. For the case of $d=2$, they showed that if $|E| \gg q^{\frac{10}{3}}$, then $ |Δ_{2, 2}(E)| \gg q^2$. In this paper, we present extensions and improvements of these results.
Erdős--Falconer distance conjecture from an analytic perspective
从解析角度研究埃尔德什 - 法尔科纳距离猜想，开发半定德尔萨特框架，假设克洛斯特曼反集中猜想，证明了关于集合 \(E,F\subset \mathbb{F}_q^{2m}\) 的距离结论，还给出了不同阈值及相关论证，并为猜想提供证据和讨论局限性。
On Densities in the Erdős-Falconer Distance Problems over the Integers
本文提出整数格中的Erdős-Falconer距离问题，以离散类比、Falconer问题等为动机，用离散球面平均的ℓᵖ改进不等式研究，其方法在部分场景优于现有结果，或可助力原始固定距离问题。
The Erdős-Falconer distance problem between arbitrary sets and $k$-coordinatable sets in finite fields
论文研究有限域上任意集合与$k$-可坐标化集合间的距离集大小，证明乘积条件$|A||B|>2q^d$可保证超过$q/2$个不同距离，并恢复奇数维尖锐阈值、改进Box距离问题结果。
The first purpose of this paper is to provide new finite field extension theorems for paraboloids and spheres. By using the unusual good Fourier transform of the zero sphere in some specific dimensions, which has been discovered recently in the work of Iosevich, Lee, Shen, and the first and second listed authors (2018), we provide a new $L^2\to L^r$ extension estimate for paraboloids in dimensions $d=4k+3$ and $q\equiv 3\mod 4$, which improves significantly the recent exponent obtained by the first listed author. In the case of spheres, we introduce a way of using \textit{the first association…
In this paper we investigate three unsolved conjectures in geometric combinatorics, namely Falconer's distance set conjecture, the dimension of Furstenburg sets, and Erdos's ring conjecture. We formulate natural $δ$-discretized versions of these conjectures and show that in a certain sense that these discretized versions are equivalent. In particular, it appears that to progress on any of these problems one must prove a quantitative statement about the existence of sub-rings of $R$ of dimension 1/2.
A Delsarte Linear Programming Approach to the Erdős--Falconer Distance Problem over Finite Fields
引入Delsarte线性规划方法，证明对偶维有限域上非退化二次型，当点集大小≥C q^{n/2+1/3}时，距离集包含正比例元素，改进了Iosevich–Rudnev的指数(n+1)/2至n/2+1/3。
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 $Δ(E)=\{|x-y|: x,y \in E \}$ is positive if the Hausdorff dimension of $E$ is greater than $\frac{4}{3}$. In this paper we improve the $\frac{4}{3}$ barrier by a small exponent for Cartesian products. In higher dimensions, also in the context of Cartesian products, we reduce Erdogan's $\frac{d}{2}+\frac{1}{3}$ exponent to $\frac{d^2}{2d-1}$. The proof uses a combination of Fourier analysis and additive comibinatorics.
Sphere intersections and incidences over finite fields
本文通过估计球面两两交集的点数之和，建立有限向量空间中点与球面的新关联界，并给出奇数维 Erdős-Falconer 距离问题中 Iosevich-Rudnev 界的初等证明。
We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving $(k+1)$-point configurations in geometric measure theory, with $k \ge 2$, including the distribution of simplices, volumes and angles determined by the points of fractal subsets $E \subset {\Bbb R}^d$, $d \ge 2$. If $T_k(E)$ denotes the set of noncongruent $(k+1)$-point configurations determined by $E$, we show that if the Hausdorff dimension of $E$ is greater than $d-\frac{d-1}{2k}$, then th…
Parabolic distance in $\mathbb F_q^2$: a sharp exponent and new results
本文研究有限域中抛物线距离问题，确定子集决定多个不同抛物线距离的阈值，结合傅里叶分析和加性组合学方法，提供正向结果及精确性示例。
We study the Erd\H os-Falconer distance problem for a set $A\subset \mathbb{F}^2$, where $\mathbb{F}$ is a field of positive characteristic $p$. If $\mathbb{F}=\mathbb{F}_p$ and the cardinality $|A|$ exceeds $p^{5/4}$, we prove that $A$ determines an asymptotically full proportion of the feasible $p$ distances. For small sets $A$, namely when $|A|\leq p^{4/3}$ over any $\mathbb{F}$, we prove that either $A$ determines $\gg|A|^{2/3}$. For both large and small sets, the results proved are in fact for pinned distances.
Here we examine some Erdos-Falconer-type problems in vector spaces over finite fields involving right angles. Our main goals are to show that a) a subset A of F_q^d of size >> q^[(d+2)/3] contains three points which generate a right angle, and b) a subset A of F_q^d of size >> q^[(d+2)/2] contains two points which generate a right angle with the vertex at the origin. We will also prove that b) is sharp up to constants and provide some partial results for similar problems related to spread and collinear triples.
In this paper we investigate the Erdös/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 classically used to study this problem. We conjecture that a majorant for the spherical means suffices to prove the distance conjecture(s) in this setting. For a class of non-Euclidean distances, we show that this generally cannot be achieved, at least in dimension two, by considering integer point distributions on convex curves and surfaces. In higher dimensions…
We study the Erdos distance conjecture on the unit sphere in three dimensions using Fourier analytic methods.
Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in 3-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite non-ovoidal circle geometries. Using the same techniques, we show a stability…
In this paper, we study the Erdős-Falconer distance problem in five dimensions for sets of Cartesian product structures. More precisely, we show that for $A\subset \mathbb{F}_p$ with $|A|\gg p^{\frac{13}{22}}$, then $Δ(A^5)=\mathbb{F}_p$. When $|A-A|\sim |A|$, we obtain stronger statements as follows: If $|A|\gg p^{\frac{13}{22}}$, then $(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p.$ If $|A|\gg p^{\frac{4}{7}}$, then $(A-A)^2+(A-A)^2+A^2+A^2+A^2+A^2=\mathbb{F}_p.$ We also prove that if $p^{4/7}\ll |A-A|=K|A|\le p^{5/8}$, then \[|A^2+A^2|\gg \min \left\lbrace \frac{p}{K^4}, \frac{|A|^{8/3}}{K^{7/3}p^{2…
VC-dimension of Salem sets over finite fields
本文研究了有限域上Salem集的VC维数，通过扩展现有技术，提出了统一框架以分析结构化集合和随机集合。
We study the $k$-resultant modulus set problem in the $d$-dimensional vector space $\mathbb F_q^d$ over the finite field $\mathbb F_q$ with $q$ elements. Given $E\subset \mathbb F_q^d$ and an integer $k\ge 2$, the $k$-resultant modulus set, denoted by $Δ_k(E)$, is defined as $$ Δ_k(E)=\{\|x^1\pm x^2 \pm \cdots \pm x^k\|\in \mathbb F_q: x^j\in E, ~j=1,2,\ldots, k\},$$ where $\|α\|=α_1^2+\cdots+ α_d^2$ for $α=(α_1, \ldots, α_d) \in \mathbb F_q^d.$ In this setting, the $k$-resultant modulus set problem is to determine the minimal cardinality of $E\subset \mathbb F_q^d$ such that $Δ_k(E) = \mathbb…
Let $\mathbb{F}_q$ be a finite field of order $q$ and $\mathcal{E}$ be a set in $\mathbb{F}_q^d$. The distance set of $\mathcal{E}$, denoted by $Δ(\mathcal{E})$, is the set of distinct distances determined by the pairs of points in $\mathcal{E}$. Very recently, Iosevich, Koh, and Parshall (2018) proved that if $|\mathcal{E}|\gg q^{d/2}$, then the quotient set of $Δ(\mathcal{E})$ satisfies \[\left\vert\frac{Δ(\mathcal{E})}{Δ(\mathcal{E})}\right\vert=\left\vert \left\lbrace\frac{a}{b}\colon a, b\in Δ(\mathcal{E}), b\ne 0\right\rbrace\right\vert\gg q.\] In this paper, we break the exponent $d/2$ …
A celebrated unit distance conjecture due to Erd\H os says that that the unit distances cannot arise more than $C_εn^{1+ε}$ times (for any $ε>0$) among $n$ points in the Euclidean plane (see e.g. \cite{SST84} and the references contained therein). In three dimensions, the conjectured bound is $Cn^{\frac{4}{3}}$ (see e.g. \cite{KMSS12} and \cite{Z19}). In dimensions four and higher, this problem, in its general formulation, loses meaning because the Lens example shows that one can construct a set of $n$ points in dimension $4$ and higher where the unit distance arises $\approx n^2$ times (see e…
A recent generalization of the Erdős Unit Distance Problem, proposed by Palsson, Senger and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in $3$-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius $1/\sqrt{2}$. We also consider a similar problem about $3$-regular unit distance graphs in $\mathbb{R}^3$.
Intersection patterns and connections to distance problems
本文研究了有限域中交集现象与距离问题之间的联系，证明了在正交变换下交集大小的下界及距离问题的指数增长特性。
On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications
本文引入水平切片和平面-直线停止时间分解方法，通过将每个临界水平切片分解为富平面、富直线-贫平面和贫直线-贫平面分量，克服了球面Bochner-Riesz核中的Kloosterman障碍，证明了对于每个非零球面S_j和每个r>23/7，有R_{S_j}^*(2→r)≲1，从而在四维素数域Erdős-Falconer距离问题中首次改进了二十年前的(d+1)/2阈值。
Clique complexes of Erdős-Rényi random graphs with edge probability between $n^{-{1\over 3}}$ and $n^{-{1\over 2}}$ are shown to be aas not simply connected. This entails showing that a connected two dimensional simplicial complex for which every subcomplex has fewer than three times as many edges as vertices must have the homotopy type of a wedge of circles, two spheres and real projective planes. Note that $n^{-{1\over 3}}$ is a threshold for simple connectivity and $n^{-{1\over 2}}$ is one for vanishing first $\F_2$ homology.
Large grid subsets without many cospherical points
受射影代数几何启发，构造大小为 n-o(n) 的 d 维网格子集，避免 d+2 个点共球或共超平面，改进 Erdős--Purdy 问题下界并渐近解决广义版本。
Quadratic distances in even dimensions over prime fields
本文研究素数域上偶数维中非退化二次型诱导的距离集大小，证明了当集合大小达到 $p^m$ 时距离集下界为 $p/\log(2+p^{m+1}/|E|)$，并在平面情形给出最优（对数因子内）下界 $|E|/\log(2|E|)$。
In the random geometric graph model $\mathsf{Geo}_d(n,p)$, we identify each of our $n$ vertices with an independently and uniformly sampled vector from the $d$-dimensional unit sphere, and we connect pairs of vertices whose vectors are ``sufficiently close'', such that the marginal probability of an edge is $p$. We investigate the problem of testing for this latent geometry, or in other words, distinguishing an Erdős-Rényi graph $\mathsf{G}(n, p)$ from a random geometric graph $\mathsf{Geo}_d(n, p)$. It is not too difficult to show that if $d\to \infty$ while $n$ is held fixed, the two distrib…
Congruence Classes of Simplex Structures in Finite Field Vector Spaces
研究有限域上 Erdős-Falconer 距离问题的推广，提出分支平移和单纯形失衡的几何技术，建立处理单纯形树等图结构的统一框架，获得了新的非平凡界并改进了现有结果。
We discuss a unified approach to a class of geometric combinatorics incidence problems in $2D$, of the Erdös distance type. The goal is obtaining the second moment estimate, that is given a finite point set $S$ and a function $f$ on $S\times S$, an upper bound on the number of solutions of $$ f(p,p') = f(q,q')\neq 0,\qquad (p,p',q,q')\in S\times S\times S\times S. \qquad(*) $$ E.g., $f$ is the Euclidean distance in the plane, sphere, or a sheet of the two-sheeted hyperboloid. Our tool is the Guth-Katz incidence theorem for lines in $\mathbb{RP}^3$, but we focus on how the original $2D$ problem…
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