A. Iosevich, I. Laba
We prove a series of results on the size of distance sets corresponding to sets in the Euclidean space. These distances are generated by bounded convex sets and the results depend explicitly on the geometry of these sets. We also use a diophantine mechanism to convert continuous results into distance set estimates for discrete point sets.
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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 prove some weighted Fourier restriction estimates using polynomial partitioning and refined Strichartz estimates. As application we obtain improved spherical average decay rates of the Fourier transform of fractal measures, and therefore improve the results for the Falconer distance set conjecture in three and higher dimensions.
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.
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 Lebesgue measure as soon the Hausdorff dimension of $E$ is larger than $n/2$. The best results in this direction have exploited estimates that restrict the Fourier transform of measures to the $(n-1)$-dimensional sphere. Here we show that these estimates can be replaced by estimates that restrict the Fourier transform of measures to the $n$-dimensional cone. Such…
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 best known result towards Falconer's distance set conjecture in even dimensions.
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。
Let $S \subset {\mathbb R}^d$ be contained in the unit ball. Let $Δ(S)=\{||a-b||:a,b \in S\}$, the Euclidean distance set of $S$. Falconer conjectured that the $Δ(S)$ has positive Lebesque measure if the Hausdorff dimension of $S$ is greater than $\frac{d}{2}$. He also produced an example, based on the integer lattice, showing that the exponent $\frac{d}{2}$ cannot be improved. In this paper we prove the Falconer distance conjecture for this class of sets based on the integer lattice. In dimensions four and higher we attain the endpoint by proving that the Lebesgue measure of the resulting dis…
Slices and distances: on two problems of Furstenberg and Falconer
本文综述了遍历论中Furstenberg的×2、×3切片猜想与几何测度论中Falconer距离集问题的研究历史与最新进展，介绍了相关求解思路，并指出二者可通过分支数组合分析、多尺度投影视角的共通方法开展研究。
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 prove, for all $d \ge 2$, that for no $s<\frac{d+1}{2}$ does $I_s(μ)<\infty$ imply the canonical Falconer distance problem incidence bound, or the analogous estimate where the Euclidean norm is replaced by the norm generated by a particular convex body $B$ with a smooth boundary and everywhere non-vanishing curvature. Our construction, based on a combinatorial construction due to Pavel Valtr naturally leads us to some interesting connections between the problem under consideration, geometric incidence theorem in the discrete setting and distribution of lattice points in convex…
On Falconer type functions and the distance set problem
本文研究了Falconer型函数的距离集问题，证明了在特定条件下距离集具有正测度，并扩展了对光滑函数的分析，恢复了关于二次多项式的尖锐结果。
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。
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…
Falconer's problem for dot product on paraboloids
本文研究了抛物面紧子集的点积Falconer问题，通过将其重新表述为距离集变体并利用固定距离问题的最新成果，证明了在特定维数阈值下点积集具有正Lebesgue测度，从而降低了传统维数障碍。
On volume vectors determined by hypergraphs in thin subsets of Euclidean space
本文针对Falconer距离问题的三点变体，研究三角形面积和单形体体积，提出了雅可比方法和基于Shmerkin-Yavicoli工作的精细方法，在高维和平面情形下分别获得了体积向量的非平凡维数阈值。
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.
We study the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. Key new ingredients include a family of refined microlocal decoupling inequalities, which are related to the work of Beltran-Hickman-Sogge on Wolff-type inequalities, and an analog of Orponen's radial projection lemma which has proved quite useful in recent work on distance sets.
We prove that if $A$ is a Borel set in the plane of equal Hausdorff and packing dimension $s>1$, then the set of pinned distances $\{ |x-y|:y\in A\}$ has full Hausdorff dimension for all $x$ outside of a set of Hausdorff dimension $1$ (in particular, for many $x\in A$). This verifies a strong variant of Falconer's distance set conjecture for sets of equal Hausdorff and packing dimension, outside the endpoint $s=1$.
On the number of 3APs in fractal sets
本文研究 fractal 集合中三元等差数列的步长长度集合的大小问题，提出基于测度质量下界和傅里叶变换范数上界的新结果。
An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $α>0$ such that $|Δ(E)| \gtrsim q$ whenever $|E| \gtrsim q^α$, where $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $Δ(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in E\}$. In two dimensions we improve the known exponent to $\tfrac{4}{3}$, consistent with the corresponding exponent in Euclidean space obtained by Wolff. The pinned distance set $Δ_y(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x\in E\}$ for a pin $y\in E$ has…
We consider the Assouad dimension analogues of two important problems in geometric measure theory. These problems are tied together by the common theme of `passing to weak tangents'. First, we solve an analogue of Falconer's distance set problem for Assouad dimension in the plane: if a planar set has Assouad dimension strictly greater than 1, then its distance set has Assouad dimension 1. We also obtain partial results in higher dimensions. Second, we consider how Assouad dimension behaves under orthogonal projection. We extend the planar projection theorem of Fraser and Orponen to higher dime…
Limits of Discrete Energy of Families of Increasing Sets
研究点序列填充集合时，探讨其里斯能量离散类似物对集合豪斯多夫维数的界定，还涉及在数据科学及相关问题中的应用。
A probabilistic analogue of the Fourier extension conjecture
本文通过概率方法证明了傅里叶扩展定理，利用平滑的Alpert乘子进行平均，结合 stationary phase 和 L^2/L^4 插值技术，得出了共振形式的正确 L^4 估计。
We prove $d$-linear analogues of the classical restriction and Kakeya conjectures in $\R^d$. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called "joints" problem, as well as presenting some $n$-linear analogues for $n<d$.
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。
A conjecture of I. Krasikov is proved. Several discrete analogues of classical polynomial inequalities are derived, along with results which allow extensions to a class of transcendental entire functions in the Laguerre-Pólya class.
An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold $α>0$ such that $|Δ(E)| \gtrsim q$ whenever $|E| \gtrsim q^α$, where $E \subset {\Bbb F}_q^d$, the $d$-dimensional vector space over a finite field with $q$ elements (not necessarily prime). Here $Δ(E)=\{{(x_1-y_1)}^2+...+{(x_d-y_d)}^2: x,y \in E\}$. The second listed author and Misha Rudnev established the threshold $\frac{d+1}{2}$, and the authors of this paper, Doowon Koh and Misha Rudnev proved that this exponent is sharp in even dimensions. In this paper we improve the threshold to $\frac…
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…
On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$
该研究在全维数下对满足Hausdorff与packing维数相等的集合证明了Falconer距离集猜想的维数版本，给出低维及高维下距离集维数的显式估计，依赖的径向投影维新估计具独立价值。
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