arXiv subjects

Nets Katz

Publications and source records attributed to Nets Katz.

6 recordsLinked to original sources

arXiv2023

Structure of cell decompositions in Extremal Szemer\'edi-Trotter examples

The symmetric case of the Szemer\'edi-Trotter theorem says that any configuration of $N$ lines and $N$ points in the plane has at most $O(N^{4/3})$ incidences. We describe a recipe involving just $O(N^{1/3})$ parameters which sometimes (that is, for some choices of the parameters) produces a configuration of N point and N lines. (Otherwise, we say the recipe fails.) We show that any near-extremal example for Szemer\'edi Trotter is densely related to a successful instance of the recipe. We obtain this result by getting structural information on cell decompositions for extremal Szemer\'edi-Trotter examples. We obtain analogous results for unit circles.

math.CO↗

arXiv2006

A sum-product estimate in finite fields, and applications

Let $A$ be a subset of a finite field $F := \Z/q\Z$ for some prime $q$. If $|F|^δ< |A| < |F|^{1-δ}$ for some $δ> 0$, then we prove the estimate $|A+A| + |A.A| \geq c(δ) |A|^{1+\eps}$ for some $\eps = \eps(δ) > 0$. This is a finite field analogue of a result of Erdos and Szemeredi. We then use this estimate to prove a Szemeredi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdos distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.

math.CO↗

arXiv2001

Fourier bases and a distance problem of Erd\H os

We prove that no ball admits a non-harmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erd\H os, which says that the number of distances determined by $n$ points in ${\Bbb R}^d$ is at least $C_d n^{\frac{1}{d}+ε_d}$, $ε_d>0$.

math.CA↗

arXiv2001

Fuglede conjecture holds for convex planar domains

Let $Ω$ be a compact convex domain in the plane. We prove that $L^2(Ω)$ has an orthogonal basis of exponentials if and only if $Ω$ tiles the plane by translation.

math.CA↗

arXiv2001

New bounds on Kakeya problems

We establish new estimates on the Minkowski and Hausdorff dimensions of Besicovitch sets and obtain new bounds on the Kakeya maximal operator.

math.CA↗

arXiv2000

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).

math.CA↗