Related papers: Conical Kakeya and Nikodym Sets in Finite Fields

We prove the joints conjecture, showing that for any $N$ lines in ${\Bbb R}^3$, there are at most $O(N^{{3 \over 2}})$ points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$ lines…

We study some systems of polynomials whose support lies in the convex hull of a circuit, giving a sharp upper bound for their numbers of real solutions. This upper bound is non-trivial in that it is smaller than either the Kouchnirenko or…

$K_\sigma$ sets involving sticky maps $\sigma$ have been used in the theory of differentiation of integrals to probabilistically construct Kakeya-type sets that imply certain types of directional maximal operators are unbounded on…

If the non-commutative L p space of SLn(Z) has the completely bounded approximation property for some non-trivial value of p, then some form of the Kakeya conjecture holds in dimension d, for all d $\le$ n+1 2 . The proof relies on a…

We study several distinct but related Fourier analytic variants of the well-known Kakeya and Furstenberg set problems in the plane. For example, given $0<s,t<1$, we call a set $K \subseteq \mathbb{R}^2$ an $(s,t)$-Kakeya set if there exists…

Around the early 2000-s, Bourgain, Katz and Tao introduced an arithmetic approach to study Kakeya-type problems. They showed that the Euclidean Kakeya conjecture follows from a natural problem in additive combinatorics, now referred to as…

We develop a notion of finite order lacunarity for direction sets in $\mathbb R^{d+1}$. Given a direction set $\Omega$ that is sublacunary according to this definition, we construct random examples of Euclidean sets that contain unit line…

In this paper, we study continuous Kakeya line and needle configurations, of both the oriented and unoriented varieties, in connected Lie groups and some associated homogenous spaces. These are the analogs of Kakeya line (needle) sets…

This paper develops an information-theoretic framework for algorithmic complexity under regular identifiable fibering. The central question is: when a decoder is given information about the fiber label in a fibered geometric set, how much…

We show that the problem of counting collinear points in a permutation (previously considered by the author and J. Solymosi in "Collinear Points in Permutations", 2005) and the well-known finite plane Kakeya problem are intimately…

A subset U of a group G is called k-universal if U contains a translate of every k-element subset of G. We give several nearly optimal constructions of small k-universal sets, and use them to resolve an old question of Erdos and Newman on…

This small note presents in any dimension a family of cubics that have finite-dimensional motive (in the sense of Kimura). As an illustration, we verify a conjecture of Voevodsky for these cubics, and a conjecture of Murre for the Fano…

We prove that a subset of $\mathbb{F}_q^n$ that contains a hyperplane in any direction has size at least $q^{n}-O(q^2)$.

We give new lower bounds for the Hausdorff dimension of Kakeya sets built from various families of curves in $\mathbb R^3$, going beyond what the polynomial partitioning method has so-far achieved. We do this by combining Wolff's classical…

We construct a compact set in $\mathbb R^2$ of measure 0 containing a piece of a parabola of every aperture between 1 and 2. As a consequence, we improve lower bounds for the $L^p$-$L^q$ norm of the corresponding maximal operator for a…

Let $v$ be a finite place of a number field $K$ and write $K^{nr,v}$ for the maximal field extension of $K$ in which $v$ is unramified. The purpose of this paper is split up into two parts. The first one generalizes a theorem of Pottmeyer:…

We introduce the theory of div point sets, which aims to provide a framework to study the combinatoric nature of any set of points in general position on an Euclidean plane. We then show that proving the unsatisfiability of some first-order…

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…

A Sidon set is a set of integers containing no nontrivial solutions to the equation $a+b=c+d$. We improve on the lower bound on the diameter of a Sidon set with $k$ elements: if $k$ is sufficiently large and ${\cal A}$ is a Sidon set with…

Efficient algorithms for convex optimization, such as the ellipsoid method, require an a priori bound on the radius of a ball around the origin guaranteed to contain an optimal solution if one exists. For linear and convex quadratic…