7 papers · 1 filter
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…