108 citations · 378 across the 31 of their papers we have counts for
16 papers · 1 filter
Kakeya sets from lines in
Nets Hawk Katz, Shukun Wu, Joshua Zahl
We prove that every Kakeya set in formed from lines of the form with must have Hausdorff dimension ; Kakeya sets…
A Kakeya maximal function estimate in four dimensions using planebrushes
Nets Hawk Katz, Joshua Zahl
We obtain an improved Kakeya maximal function estimate and improved Kakeya Hausdorff dimension estimate in using a new geometric argument called the planebrush. A pl…
On the discretized sum-product problem
Larry Guth, Nets Hawk Katz, Joshua Zahl
We give a new proof of the discretized ring theorem for sets of real numbers. As a special case, we show that if is a -set in the sense of Katz and…
On the polynomial Wolff axioms
Nets Hawk Katz, Keith M. Rogers
We confirm a conjecture of Guth concerning the maximal number of -tubes, with -separated directions, contained in the -neighborhood of a real algebraic variety. Modulo a f…
An improved bound on the Hausdorff dimension of Besicovitch sets in
Nets Hawk Katz, Joshua Zahl
We prove that any Besicovitch set in must have Hausdorff dimension at least for some small constant . This follows from a more general result about…
New Bounds on cap sets
Michael Bateman, Nets Hawk Katz
We provide an improvement over Meshulam's bound on cap sets in . We show that there exist universal and so that any cap set in has size at most $C {3^N \…
Kakeya Sets in Cantor directions
Michael D. Bateman, Nets Hawk Katz
We construct a union of N parallelograms of dimensions approximately 1/N x 1 in the plane, with the slope of their long sides in the standard Cantor set. The union has area 1/log N…
On Sums, Products, and the multidimensional Falconer problem
Nets Hawk Katz
A set in d dimensional Euclidean space with d larger than 2 having Hausdorff dimension at least d/2 must have distance set with Hausdorff dimension strictly greater than 1/2.
The Grone Merris Conjecture and a quadratic eigenvalue problem
Nets Hawk Katz
We verify the Grone Merris conjecture for a class of graphs. We do this by curve sketching in the sense of first year calculus. That is, we do it by homotopy methods.
Fourier bases and a distance problem of Erd\H os
Alex Iosevich, Nets Katz, Steen Pedersen
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…
Fuglede conjecture holds for convex planar domains
Alex Iosevich, Nets Katz, Terry Tao
Let be a compact convex domain in the plane. We prove that has an orthogonal basis of exponentials if and only if tiles the plane by translation.
New bounds on Kakeya problems
Nets Katz, Terence Tao
We establish new estimates on the Minkowski and Hausdorff dimensions of Besicovitch sets and obtain new bounds on the Kakeya maximal operator.
Some connections between Falconer's distance set conjecture, and sets of Furstenburg type
Nets Hawk Katz, Terence Tao
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 c…
Recent progress on the Kakeya conjecture
Nets Katz, Terence Tao
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…
Convex bodies with a point of curvature do not have Fourier bases
Alex Iosevich, Nets Hawk Katz, Terence Tao
We prove that no smooth symmetric convex body with at least one point of non-vanishing Gaussian curvature can admit an orthogonal basis of exponentials. (The non-symmetric case…
An improved bound on the Minkowski dimension of Besicovitch sets in R^3
Nets Hawk Katz, Izabella Łaba, Terence Tao
A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal…