New bounds on Kakeya problems

R⁴中Kakeya集的Hausdorff维数新下界

New bound on the Hausdorff dimension of Kakeya sets in $\mathbb{R}^4$

该研究将Wang-Zahl的三维Kakeya集研究与Katz-Zahl的平面刷论证结合，改进Guth-Zahl的三线性界，得到R⁴中Kakeya集Hausdorff维数的新下界3.073。

Kakeya Sets and Directional Maximal Operators in the Plane

We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets, then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p, for p>1.

$\nmathbb{R}^4$ 中限制性与 Kakeya 问题范围的改进

Refinements on the Restriction and Kakeya problems range in $\mathbb{R}^4$

通过证明两个两端点关联估计，改进了 $\mathbb{R}^4$ 中 Fourier 限制性与 Kakeya 极大估计的范围，并探讨了对 Kakeya 集维数下界和 Bochner--Riesz 猜想的影响。

A recursive bound for a Kakeya-type maximal operator

A (d,k) set is a subset of R^d containing a translate of every k-dimensional plane. Bourgain showed that for 2^{k-1}+k \geq d, every (d,k) set has positive Lebesgue measure. We give an L^p bound for the corresponding maximal operator.

The Kakeya needle problem and the existence of Besicovitch and Nikodym sets for rectifiable sets

We solve the Kakeya needle problem and construct a Besicovitch and a Nikodym set for rectifiable sets.

$\u211d^4$中粘性Kakeya集Hausdorff维数的一个改进界

An improved bound on the Hausdorff dimension of sticky Kakeya sets in $\mathbb{R}^4$

本文针对$\u211d^4$中的粘性Kakeya集，结合平面刷论证与粘性性质，将其Hausdorff维数下界从原有3.059提升至3.25，推进了粘性Kakeya猜想的相关研究。

$L^2$ bounds for a Kakeya type maximal operator in $\R^3$

We prove that the maximal operator obtained by taking averages at scale 1 along $N$ arbitrary directions on the sphere, is bounded in $L^2(\R^3)$ by $N^{1/4}{\log N}$. When the directions are $N^{-1/2}$ separated, we improve the bound to $N^{1/4}\sqrt{\log N}$. Apart from the logarithmic terms these bounds are optimal.

Improved bounds for the Kakeya maximal conjecture in higher dimensions

We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.

On the size of Kakeya sets in finite fields

A Kakeya set is a subset of F^n, where F is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least C_n * q^n, where C_n depends only on n. This improves the previously best lower bound for general n of ~q^{4n/7}.

关于 Peres--Schlag 正交投影问题和 Kakeya 型集合

On the Peres--Schlag orthogonal projection problem and Kakeya-type sets

研究有限域和欧几里得空间中正交投影的 Peres--Schlag 非空内部问题，通过多项式方法建立有限域上的尖锐投影结果，并在欧几里得空间中结合几何测度论与调和分析技术改进参数范围。

The Fourier restriction and Kakeya problems over rings of integers modulo $N$

The Fourier restriction phenomenon and the size of Kakeya sets are explored in the setting of the ring of integers modulo $N$ for general $N$ and a striking similarity with the corresponding euclidean problems is observed. One should contrast this with known results in the finite field setting.

Sharp bounds for multilinear curved Kakeya, restriction and oscillatory integral estimates away from the endpoint

We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates that were obtained in paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form $R^\varepsilon$ (or $\log^{O(1)} R$) for some scale factor $R$. By further developing the heat flow method, and applying it directly for the first time to the multilinear curved Kakeya and restriction settings, we are able to eliminate these losses, as long a…

Recent progress on the Kakeya conjecture

The purpose of this article is to survey the developments on the Kakeya problem in recent years, concentrating on the period after the excellent 1999 survey of Wolff, and including some recent work by the authors. We will focus on the standard Kakeya problem for line segments and not discuss other important variants (such as Kakeya estimates for circles, light rays, or $k$-planes).

Refined and microlocal Kakeya-Nikodym bounds for eigenfunctions in two dimensions

We obtain some improved essentially sharp Kakeya-Nikodym estimates for eigenfunctions in two-dimensions. We obtain these by proving stronger related microlocal estimates involving a natural decomposition of phase space that is adapted to the geodesic flow.

Quasisymmetric maps on Kakeya sets

I show that $L^{p}-L^{q}$ estimates for the Kakeya maximal function yield lower bounds for the conformal dimension of Kakeya sets, and upper bounds for how much quasisymmetries can increase the Hausdorff dimension of line segments inside Kakeya sets. Combining the known $L^{p}-L^{q}$ estimates of Wolff and Katz-Tao with the main result of the paper, the conformal dimension of Kakeya sets in $\mathbb{R}^{n}$ is at least $\max\{(n + 2)/2,(4n + 3)/7\}$. Moreover, if $f$ is a quasisymmetry from a Kakeya set $K \subset \mathbb{R}^{n}$ onto any at most $n$-dimensional metric space, the $f$-image of …

Restriction and Kakeya phenomena for finite fields

The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and are related to many problems in harmonic analysis, PDE, and number theory. In this paper we initiate the study of these problems on finite fields. The restriction problem then becomes a question about general bounds for certain types of exponential sums, while the Kakeya problem has an algebraic geometry flavor, asking for the extent to which lines in different directions can overlap. In many cases the Euclidean arguments carry over easily to the finite setting (and are in fact some…

几乎不满足反向Kakeya条件的可实现Cantorval

Achievable Cantorvals almost without reversed Kakeya conditions

该研究构造了仅在渐近密度为零的集合上满足反向Kakeya条件的可实现Cantorval，正面解答了相关公开问题，同时发现其边界Lebesgue测度为零，未解决边界正测度可实现Cantorval的存在性问题。

有限域中挂川集合的 Fourier 分析性质

Fourier analytic properties of Kakeya sets in finite fields

本文利用 Fourier 分析证明有限域挂川集合可支撑非零频率变换受 $q^{-1}$ 控制的概率测度，并给出二维下界的新证明及 $k$ 平面推广。

New Kakeya estimates using the polynomial Wolff axioms

We obtain new bounds for the Kakeya maximal conjecture in most dimensions $n<100$, as well as improved bounds for the Kakeya set conjecture when $n=7$ or $9$. For this we consider Guth and Zahl's strengthened formulation of the maximal conjecture, concerning families of tubes that satisfy the polynomial Wolff axioms. Our results give improved estimates for this strengthened formulation when $n =5$ or $n \geq 7$.

On the Dimension of Kakeya Sets in the First Heisenberg Group

We define Kakeya sets in the Heisenberg group and show that the Heisenberg Hausdorff dimension of Kakeya sets in the first Heisenberg group is at least 3. This lower bound is sharp since, under our definition, the $\{xoy\}$-plane is a Kakeya set with Heisenberg Hausdorff dimension 3.

星形Kakeya集的下界改进

An improved lower bound for star-shaped Kakeya sets

本文改进了Cunningham关于星形Kakeya集面积的下界，从π/108提升至π/98。

Kakeya 猜想证明导论

Introduction to the proof of the Kakeya conjecture

本文综述 Hong Wang 与 Joshua Zahl 对三维 Kakeya 猜想的证明，梳理问题背景、既有工作、核心困难与新方法。

粘性Kakeya集与粘性Kakeya猜想

Sticky Kakeya sets and the sticky Kakeya conjecture

本文提出并证明了粘性Kakeya集的猜想，表明其Hausdorff和Minkowski维度为n。

粘性管集的Kakeya与Furstenberg问题

Kakeya and Furstenberg problems for sticky sets of tubes

本文研究粘性管集的关联问题，证明了三维空间中满足凸Wolff公理的粘性管集的尖锐Furstenberg集估计，以及四维空间中相应的Kakeya集猜想。

Refined and Microlocal Kakeya-Nikodym Bounds of Eigenfunctions in Higher Dimensions

We prove a Kakeya-Nikodym bound on eigenfunctions and quasimodes, which sharpens a result of the authors and extends it to higher dimensions. As in the prior work, the key intermediate step is to prove a microlocal version of these estimates, which involves a phase space decomposition of these modes which is essentially invariant under the bicharacteristic/geodesic flow. In a companion paper, it will be seen that these sharpened estimates yield improved $L^q(M)$ bounds on eigenfunctions in the presence of nonpositive curvature when $2 < q < \frac{2(d+1)}{d-1}$.

The Kakeya problem for circular arcs

We prove that if a circular arc has angle short enough, then it can be continuously moved to any prescribed position within a set of arbitrarily small area.

矩阵多项式的 Enestrom-Kakeya 型定理推广

Generalizations of Enestrom-Kakeya Type Theorems for matrix Polynomials

该研究针对矩阵多项式的特征值定位问题，推广经典 Enestrom-Kakeya 定理，给出多种适用于矩阵多项式的特征值界。

On the Multilinear Restriction and Kakeya conjectures

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 $n<d$.

Dimension estimates for Kakeya sets defined in an axiomatic setting

In this dissertation we define a generalization of Kakeya sets in certain metric spaces. Kakeya sets in Euclidean spaces are sets of zero Lebesgue measure containing a segment of length one in every direction. A famous conjecture, known as Kakeya conjecture, states that the Hausdorff dimension of any Kakeya set should equal the dimension of the space. It was proved only in the plane, whereas in higher dimensions both geometric and arithmetic combinatorial methods were used to obtain partial results. In the first part of the thesis we define generalized Kakeya sets in metric spaces satisfying c…

Bounds for Kakeya-type maximal operators associated with k-planes

A (d,k) set is a subset of R^d containing a translate of every k-dimensional plane. Bourgain showed that for k \geq k_{cr}(d), where k_{cr}(d) solves 2^{k_{cr}-1}+k_{cr} = d, every (d,k) set has positive Lebesgue measure. We give a short proof of this result which allows for an improved L^p estimate of the corresponding maximal operator, and which demonstrates that a lower value of k_{cr} could be obtained if improved mixed-norm estimates for the x-ray transform were known.

Restricting directions for Kakeya sets

We prove that the Kakeya maximal conjecture is equivalent to the $Ω$-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Math{é} proved that the Kakeya conjecture is equivalent to the $Ω$-Kakeya conjecture. Moreover, we improve concrete bound on the Hausdorff dimension of a $Ω$-Kakeya set : for any Bore set $Ω$ in S n--1 , we prove that if X $\subset$ R n contains for any e $\in$ $Ω$ a unit segment oriented along e then we have dX $\ge$ 6 11 d$Ω$ + 1 where dE denotes the Hausdorff dimension of a set E.

Grassmann流形中$k$-测地线的Kakeya问题与投影问题

Kakeya problem and projection problem for $k$-geodesics in Grassmannians

本文将Kakeya问题与投影问题推广到齐性空间，以$k$维全测地子流形替代$k$-平面，并针对Grassmann流形证明了尖锐估计。

Polynomials with dense zero sets and discrete models of the Kakeya conjecture and the Furstenberg set problem

We prove the discrete analogue of Kakeya conjecture over $\mathbb{R}^n$. This result suggests that a (hypothetically) low dimensional Kakeya set cannot be constructed directly from discrete configurations. We also prove a generalization which completely solves the discrete analogue of the Furstenberg set problem in all dimensions. The difference between our theorems and the (true) problems is only the (still difficult) issue of continuity since no transversality-at-incidences assumptions are imposed. The main tool of the proof is a theorem of Wongkew \cite{wongkew2003volumes} which states that…

Kakeya sets of curves

We investigate analogues for curves of the Kakeya problem for straight lines. These arise from H"ormander-type oscillatory integrals in the same way as the straight line case comes from the restriction and Bochner-Riesz problems. Using some of the geometric and arithmetic techniques developed for the straight line case by Bourgain, Wolff, Katz and Tao, we are able to prove positive results for families of parabolas whose coefficients satisfy certain algebraic conditions.

Discrete analogues of Kakeya problems

This thesis investigates two problems that are discrete analogues of two harmonic analytic problems which lie in the heart of research in the field. More specifically, we consider discrete analogues of the maximal Kakeya operator conjecture and of the recently solved endpoint multilinear Kakeya problem, by effectively shrinking the tubes involved in these problems to lines, thus giving rise to the problems of counting joints and multijoints with multiplicities. In fact, we effectively show that, in $\mathbb{R}^3$, what we expect to hold due to the maximal Kakeya operator conjecture, as well as…

Pattern Problems related to the Arithmetic Kakeya Conjecture

We study a variety of problems about homothets of sets related to the Kakeya conjecture. In particular, we show many of these problems are equivalent to the arithmetic Kakeya conjecture of Katz and Tao. We also provide a proof that the arithmetic Kakeya conjecture implies the Kakeya conjecture for packing dimension, as this implication was previously only known for Minkowski dimension. We consider several questions analogous to the classical results of Stein and Bourgain about the Lebesgue measure of a set containing a sphere centered at every point of $[0,1]^n$, where we replace spheres by ar…

A family of fractal Fourier restriction estimates with implications on the Kakeya problem

In a recent paper [Ann. of Math. 189 (2019), 837--861], Du and Zhang proved a fractal Fourier restriction estimate and used it to establish the sharp $L^2$ estimate on the Schrödinger maximal function in $\Bbb R^n$, $n \geq 2$. In this paper, we show that the Du-Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in $\Bbb R^n$ that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in $\Bbb R^2$.

紧李群中的Kakeya集与维数压缩

Kakeya sets and dimension compression in compact Lie groups

本文针对紧半单李群中的两类Kakeya集问题，确定了其最优Minkowski维数的上下界，推测局部Kakeya集问题的上界为精确值，结合李理论构造、关联几何与数的几何完成相关证明。

Some results in support of the Kakeya Conjecture

A Besicovitch set is a subset of $\R^d$ that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this conjecture in a variety of contexts. Our proofs are simple and aim to give an intuitive feel for the problem. For example, we give a very simple proof that the packing and lower box-counting dimension of any Besicovitch set is at least $(d+1)/2$ (better estimates are available in the literature). We also study the `generic validity' of the Kakeya conjecture in the …

On a planar variant of the Kakeya problem

A K^n_2-set is a set of zero Lebesgue measure containing a translate of every plane in an (n-2)-dimensional manifold in Gr(n,2), where the manifold fulfills a curvature condition. We show that this is a natural class of sets with respect to the Kakeya problem and prove that dim_H(E)\ge 7/2 for all K^4_2-sets E. When the underlying field is replaced by the complex numbers C, we get dim_H(E)\ge 7 for all K^4_2-sets over C, and we construct an example to show that this is sharp. Thus K^4_2-sets over C do not necessarily have full Hausdorff dimension.

Lacunarity, Kakeya-type sets and directional maximal operators

We develop a notion of finite order lacunarity for direction sets in $\mathbb R^{d+1}$. Given a direction set $Ω$ that is sublacunary according to this definition, we construct random examples of Euclidean sets that contain unit line segments with directions from $Ω$ and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. This generalizes to higher dimensions a planar result due to Bateman. Combined with earlier work of Alfonseca, Bateman, Parcet and Rogers, this notion of lacunarity and Kakeya-type sets also yields a characterization in all…

弯曲Kakeya问题与路径的射影几何

Curved Kakeya problems and the projective geometry of paths

本文为$\boldsymbol{R}^n$的弯曲Kakeya问题引入通用框架，利用喷雾几何与路径射影几何展开研究，刻画了射影平坦性与全测地曲面的关系，明确Bourgain条件的二分结果，并给出$\boldsymbol{R}^3$曲线族弯曲Kakeya集维数的等价条件。

利用格罗莫夫代数引理的新Kakeya估计

New Kakeya estimates using Gromov's algebraic lemma

该研究针对Kakeya问题，先建立方向分离管无法在低次代数簇薄邻域聚集的几何不等式，据此得到新的多线性Kakeya估计，再转化为Kakeya极大函数估计，在多数维度下得到新的界。

在R^4中的限制和凯凯ya极大估计

Restriction and Kakeya maximal estimates in $\mathbb{R}^4$

该研究结合平面刷论证与解耦- incidence方法，推导出R^4中Fourier限制和Bochner-Riesz问题的新界，并获得维度3.054的凯凯ya极大估计。

Hunter, Cauchy Rabbit, and Optimal Kakeya Sets

A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle $\Z_n$. A hunter and a rabbit move on the nodes of $\Z_n$ without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the first $n$ steps of order $1/\log n$. We show these strategies yield a Kakeya set consisting of $4n$ triangles with…

$\mathbb{R}^n$中余维二平面的Kakeya问题

The Kakeya problem for codimension-two planes in $\mathbb{R}^n$

研究n≥4时ℝⁿ中满足凸Wolff公理的余维二平面Kakeya问题，证明此类δ片的并集具有几乎最大体积，是Wang和Zahl工作的高维推广，还解决了ℝ⁴中满足曲率条件的Rogers 2平面Kakeya问题。

On Kakeya-Nikodym type maximal inequalities

We show that for any dimension $d\ge3$, one can obtain Wolff's $L^{(d+2)/2}$ bound on Kakeya-Nikodym maximal function in $\mathbb R^d$ for $d\ge3$ without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional $L^2$ estimate with an auxiliary maximal function. We also prove that the same $L^{(d+2)/2}$ bound holds for Nikodym maximal function for any manifold $(M^d,g)$ with constant curvature, which generalizes Sogge's results for $d=3$ to any $d\ge3$. As in the 3-dimensional case, we can handle manifolds of constant curvature due to the fact that, in this case, tw…

关于(n-1)可求长集的Kakeya针问题的研究

A Study on Kakeya Needle Problem for $(n-1)$-Rectifiable Sets

本文研究n维欧氏空间中针对(n-1)可求长集的Kakeya针问题类似问题，结合三类关键要素构造Nikodym型集，实现沿等距变换路径移动该类集并覆盖勒贝格零测集。

Kakeya猜想：它来自哪里以及为什么重要？

The Kakeya Conjecture: where does it come from and why is it important?

本文探讨了Kakeya猜想的起源及其在傅里叶分析中的重要性，并介绍了Hong Wang和Joshua Zahl的突破性证明。

傅里叶分析视角下的傅斯特嫩格与卡耶ya问题变体

Fourier analytic variants of the Furstenberg and Kakeya problems

本文研究了平面中著名的卡耶ya和傅斯特嫩格集合问题的几种不同的傅里叶分析变体，提出了(s,t)-卡耶ya集的概念，并推导了其傅里叶维数的上下界。

流形上的弯曲 Kakeya 集与 Nikodym 问题

Curved Kakeya sets and Nikodym problems on manifolds

本文将满足 Bourgain 条件的弯曲 Kakeya 集及常曲率流形上的 Nikodym、Furstenberg 问题归结为欧氏对应问题，从而证明三维常截面曲率流形的 Nikodym 猜想和二维常 Gauss 曲率曲面的 Furstenberg 猜想。

$L^p$-improving estimates for Radon-like operators and the Kakeya-Brascamp-Lieb inequality

This paper considers the problem of establishing $L^p$-improving inequalities for Radon-like operators in intermediate dimensions (i.e., for averages overs submanifolds which are neither curves nor hypersurfaces). Due to limitations in existing approaches, previous results in this regime are comparatively sparse and tend to require special numerical relationships between the dimension $n$ of the ambient space and the dimension $k$ of the submanifolds. This paper develops a new approach to this problem based on a continuum version of the Kakeya-Brascamp-Lieb inequality, established by Zhang and…

On Wolff's $L^{\frac{5}{2}}-$Kakeya maximal inequality in $\R^3$

We reprove Wolff's $L^{\frac{5}2}-$ bound for the $\R^3-$Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on Nikodym-type sets in curved spaces. Although the equivalence between these two type maximal functions is well known, our proof may shed light on some new geometric observations which is interesting in its own right.

任意维度下卡克亚集的构造

A construction of Kakeya Sets in Arbitrary Dimension

研究在任意维度\(d\geq 2\)下构造卡克亚集，核心方法是推广经典构造，主要贡献是所构造的卡克亚集\(\delta\)邻域体积有改进，还得出了径向傅里叶乘子\(L^p\)有界性相关结论。