46 citations · 441 across the 46 of their papers we have counts for
25 papers · 1 filter
On planar Nikodym-type sets
Alan Chang, Mingfeng Chen, Shaoming Guo +3
Nikodym constructed a measure-zero Borel set in the plane that contains a punctured line segment passing through each point in the unit square. Fenton and Falconer asked whether a…
A streamlined proof of the Kakeya set conjecture in
Larry Guth, Hong Wang, Joshua Zahl
We present a streamlined and simplified proof of the Kakeya set conjecture in .
A Survey of the Kakeya conjecture, 2000-2025
Joshua Zahl
We survey progress on the Kakeya conjecture in Euclidean space, with an emphasis on developments that have occurred since the previous surveys by Wolff and Katz-Tao.
A bound for plany Kakeya sets in using the planebrush method
Izabella Łaba, Mukul Rai Choudhuri, Joshua Zahl
Katz and Zahl used a planebrush argument to prove that Kakeya sets in have Hausdorff dimension at least 3.059. In the special case when the Kakeya set is plany, thei…
Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions
Hong Wang, Joshua Zahl
We study sets of tubes in , with the property that not too many tubes can be contained inside a common convex set . We show that the union of tubes from such a…
Improved bounds for the strong spherical maximal operator
Jonathan Hickman, Joshua Zahl
We study the mapping properties of the strong spherical maximal function, which is a multiparameter generalisation of Stein's spherical maximal function. We show that this op…
The Assouad dimension of Kakeya sets in
Hong Wang, Joshua Zahl
This paper studies the structure of Kakeya sets in . We show that for every Kakeya set , there exist well-separated scales so that…
On Maximal Functions Associated to Families of Curves in the Plane
Joshua Zahl
We consider the mapping properties of maximal averages associated to families of curves, and thickened curves, in the plane. These include the (planar) Kakeya maximal functio…
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…
Sticky Kakeya sets and the sticky Kakeya conjecture
Hong Wang, Joshua Zahl
A Kakeya set is a compact subset of that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have Hausdorff a…
Unions of lines in
Joshua Zahl
We prove a conjecture of D. Oberlin on the dimension of unions of lines in . If is an integer, , and is a set of lines in $\mathbb{R}^n…
A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in
Malabika Pramanik, Tongou Yang, Joshua Zahl
We resolve a conjecture of Fässler and Orponen on the dimension of exceptional projections to one-dimensional subspaces indexed by a space curve in . We do this by ob…
New estimates on the size of -Furstenberg sets
Daniel Di Benedetto, Joshua Zahl
We use recent advances on the discretized sum-product problem to obtain new bounds on the Hausdorff dimension of planar -Fursterberg sets. This provides a quantitative impr…
On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-Rónyai theorem
Orit E. Raz, Joshua Zahl
We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection th…
A note on Fourier restriction and nested Polynomial Wolff axioms
Jonathan Hickman, Joshua Zahl
This note records an asymptotic improvement on the known range for the Fourier restriction conjecture in high dimensions. This is obtained by combining Guth's polynomial part…
New Kakeya estimates using Gromov's algebraic lemma
Joshua Zahl
This paper presents several new results related to the Kakeya problem. First, we establish a geometric inequality which says that collections of direction-separated tubes (thin nei…
Large Sets Avoiding Rough Patterns
Jacob Denson, Malabika Pramanik, Joshua Zahl
The pattern avoidance problem seeks to construct a set with large dimension that avoids a prescribed pattern. Examples of such patterns include three-term a…
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…
A discretized Severi-type theorem with applications to harmonic analysis
Joshua Zahl
In 1901, Severi proved that if is an irreducible hypersurface in that contains a three dimensional set of lines, then is either a quadratic hyper…
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…
Polynomial Wolff axioms and Kakeya-type estimates in
Larry Guth, Joshua Zahl
We establish new linear and trilinear bounds for collections of tubes in that satisfy the polynomial Wolff axioms. In brief, a collection of -tubes satisfies the…
Quantitative visibility estimates for unrectifiable sets in the plane
M. Bond, I. Laba, J. Zahl
The "visibility" of a planar set from a point is defined as the normalized size of the radial projection of from to the unit circle centered at . Simon and Solom…
On the Wolff circular maximal function
Joshua Zahl
We prove sharp bounds for a variable coefficient generalization of the Wolff circular maximal function . For each fixed radius , is the maximal average of…