Alex Iosevich, Doowon Koh
We prove several incidence theorems in vector spaces over finite fields using bounds for various classes of exponential sums and apply these to Erdos-Falconer type distance problems.
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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.
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。
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, noncongruent point configurations of $k+1$ points from $E$) to have positive Lebesgue measure. This generalizes the $k=1$ case, the well-known Falconer distance problem and a major open problem in geometric measure theory. We establish a dimensional lower threshold of $\frac{d(k+1)}{d+2}$ for Falconer type theorems for $k$-simplices. This threshold is nontrivial i…
The Erdős-Falconer distance problem in the integer lattice
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 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…
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.
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。
We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let ${\Bbb F}_q$ be a finite field with $q$ elements and take $E \subset {\Bbb F}^d_q$, $d \ge 2$. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in ${\Bbb F}^d_q$ to provide estimates for minimum cardinality of the distance set $Δ(E)$ in terms of the cardinality of $E$. Kloosterman sums play an important role in the proof.
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…
Exploration on Incidence Geometry and Sum-Product Phenomena
本论文研究Erdős-Szemerédi猜想与关联几何的关系，将实数域上的和积估计技术推广到有限域，并在环Z/p^3Z上获得平分线数量的估计结果。
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 Erdős-Falconer distance problem: If $|A|\gg q^{\frac{d-1}{2}+\frac{1}{k-1}}$, then we have \[Δ_{k}(A)=\{|x_1+\cdots+x_k|\colon x_i\in A\}\supseteq \mathbb{F}_q^*.\] In this paper, we provide improvements and extensions of their result.
Erdős first conjectured that infinitely often we have $φ(n) = σ(m)$, where $φ$ is the Euler totient function and $σ$ is the sum of divisor function. This was proven true by Ford, Luca and Pomerance in 2010. We ask the analogous question of whether infinitely often we have $φ(F) = σ(G)$ where $F$ and $G$ are polynomials over some finite field $\mathbb{F}_q$. We find that when $q\not=2$ or $3$, then this can only trivially happen when $F=G=1$. Moreover, we give a complete characterisation of the solutions in the case $q=2$ or $3$. In particular, we show that $φ(F) = σ(G)$ infinitely often when $…
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。
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。
In this paper we obtain a new lower bound on the Erdős distinct distances problem in the plane over prime fields. More precisely, we show that for any set $A\subset \mathbb{F}_p^2$ with $|A|\le p^{7/6}$, the number of distinct distances determined by pairs of points in $A$ satisfies $$ |Δ(A)| \gg |A|^{\frac{1}{2}+\frac{149}{4214}}.$$ Our result gives a new lower bound of $|Δ{(A)}|$ in the range $|A|\le p^{1+\frac{149}{4065}}$. The main tools we employ are the energy of a set on a paraboloid due to Rudnev and Shkredov, a point-line incidence bound given by Stevens and de Zeeuw, and a lower boun…
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…
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。
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 $|E|/\log(2|E|)$。
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 $d/2$ …
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