相关论文: On the Finite Field Kakeya Problem in Two Dimensio…
In the current paper we present a new proof of the small ball inequality in two dimensions. More importantly, this new argument, based on an approach inspired by lacunary Fourier series, reveals the first formal connection between this…
Makeev conjectured that every constant-width body is inscribed in the dual difference body of a regular simplex. We prove that homologically, there are an odd number of such circumscribing bodies in dimension 3, and therefore geometrically…
We prove upper and lower bounds for leading coefficient of Kolchin dimension polynomial of systems of partial linear differential equations in codimension two.
In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of…
Let $E \subseteq \mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements, which satisfies $|E| > q$. There exist $x,y \in E$ such that $|E \cdot (y-x)| > q/2.$ In particular, $(E+E) \cdot (E-E) =…
We propose a framework for a new type of finite field theories based on a hidden duality between an ultra-violet and an infra-red region. Physical quantities do not receive radiative corrections at a fundamental scale or the fixed point of…
Motivated by the theory of graph limits, we introduce and study the convergence and limits of linear representations of finite groups over finite fields. The limit objects are infinite dimensional representations of free groups in…
There are two-dimensional Toda field equations corresponding to each (finite or affine) Lie algebra. The question addressed in this note is whether there exist integrable discrete versions of these. It is shown that for certain algebras…
The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in $\C^d.$ We study this problem on general sets, but devote special attention to product sets…
Kakeya sets are compact subsets of $\mathbb{R}^n$ that contain a unit line segment pointing in every direction. The Kakeya conjecture states that such sets must have Hausdorff dimension $n$. The property of stickiness was first discovered…
A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least…
We explore the relationship in metric spaces between different properties related to the Besicovitch covering theorem, and also consider weak versions of doubling, in connection to the non-uniqueness of centers and radii in arbitrary metric…
We consider the problem of finding the minimal number of points required to intersect all lines in an affine space over the finite field of order 3. We also consider the problem of finding the minimal number of points required to intersect…
Let $\text{PG}(n,q)$ be the Desarguesian projective space of dimension $n$ over the finite field of order $q$. The \emph{linear representation} of a point set $\mathcal{K}$ in a hyperplane at infinity of $\text{PG}(n,q)$ is the point-line…
This article is a summary of some of the author's work on Sasaki-Einstein geometry. A rather general conjecture in string theory known as the AdS/CFT correspondence relates Sasaki-Einstein geometry, in low dimensions, to superconformal…
A (1+1)-dimensional quantum field theory with a degenerate vacuum (in infinite volume) can contain particles, known as kinks, which interpolate between different vacua and have nontrivial restrictions on their multi-particle Hilbert space.…
A Kakeya set in the linear representation $T^{*}_{2}(\mathcal{C})$, $\mathcal{C}$ a non-singular conic, is the point set covered by a set of $q+1$ lines, one through each point of $\mathcal{C}$. In this article we classify the small Kakeya…
Let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements. For a subset $E\subseteq \mathbb{F}_q^d$ and a fixed nonzero $t\in \mathbb{F}_q$, let $\mathcal{H}_t(E)=\{h_y: y\in E\}$, where $h_y$ is the…
In this short note we introduce the Belyi degree of a number field K, which is the smallest degree of a dessin d'enfant having K as field of moduli. After the description of some general properties (for example, the fact that there exist…
The main result of the paper: Given any $\varepsilon>0$, every locally finite subset of $\ell_2$ admits a $(1+\varepsilon)$-bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which…