Joshua Zahl

activity

20062026

most citedSpectral gaps, additive energy, and a fractal uncertainty principle

46 citations · 441 across the 46 of their papers we have counts for

collaborators

49 papers

math.CA2026

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…

math.CA2026★ 1 cited

A streamlined proof of the Kakeya set conjecture in $R^{3}$

Larry Guth, Hong Wang, Joshua Zahl

We present a streamlined and simplified proof of the Kakeya set conjecture in $R^{3}$ .

math.CO2025

On the number of pairwise touching cylinders in $R^{d}$

Jozsef Solymosi, Josh Zahl

John E. Littlewood posted the question {\em ``Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number sugge…

math.CA2025

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.

math.CA2025

A bound for plany Kakeya sets in $F_{q}^{4}$ using the planebrush method

Izabella Łaba, Mukul Rai Choudhuri, Joshua Zahl

Katz and Zahl used a planebrush argument to prove that Kakeya sets in $R^{4}$ have Hausdorff dimension at least 3.059. In the special case when the Kakeya set is plany, thei…

math.CA2025★ 2 cited

Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions

Hong Wang, Joshua Zahl

We study sets of $δ$ tubes in $R^{3}$ , with the property that not too many tubes can be contained inside a common convex set $V$ . We show that the union of tubes from such a…

math.CA2025

Improved $L^{p}$ bounds for the strong spherical maximal operator

Jonathan Hickman, Joshua Zahl

We study the $L^{p}$ mapping properties of the strong spherical maximal function, which is a multiparameter generalisation of Stein's spherical maximal function. We show that this op…

math.CA2024★ 5 cited

The Assouad dimension of Kakeya sets in $R^{3}$

Hong Wang, Joshua Zahl

This paper studies the structure of Kakeya sets in $R^{3}$ . We show that for every Kakeya set $K \subset R^{3}$ , there exist well-separated scales $0 < δ < ρ \leq 1$ so that…

math.CA2023★ 2 cited

On Maximal Functions Associated to Families of Curves in the Plane

Joshua Zahl

We consider the $L^{p}$ mapping properties of maximal averages associated to families of curves, and thickened curves, in the plane. These include the (planar) Kakeya maximal functio…

math.CO2022★ 5 cited

Improved Elekes-Szabó type estimates using proximity

Jozsef Solymosi, Joshua Zahl

We prove a new Elekes-Szabó type estimate on the size of the intersection of a Cartesian product $A \times B \times C$ with an algebraic surface ${f = 0}$ over the reals. In particul…

math.CA2022

Kakeya sets from lines in $S L_{2}$

Nets Hawk Katz, Shukun Wu, Joshua Zahl

We prove that every Kakeya set in $R^{3}$ formed from lines of the form $(a, b, 0) + span (c, d, 1)$ with $a d - b c = 1$ must have Hausdorff dimension $3$ ; Kakeya sets…

math.CA2022★ 2 cited

Sticky Kakeya sets and the sticky Kakeya conjecture

Hong Wang, Joshua Zahl

A Kakeya set is a compact subset of $R^{n}$ that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have Hausdorff a…

math.CA2022★ 2 cited

Unions of lines in $R^{n}$

Joshua Zahl

We prove a conjecture of D. Oberlin on the dimension of unions of lines in $R^{n}$ . If $d \geq 1$ is an integer, $0 \leq β \leq 1$ , and $L$ is a set of lines in $\mathbb{R}^n…

math.CA2022★ 5 cited

A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in $R^{3}$

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 $R^{3}$ . We do this by ob…

math.CA2021★ 2 cited

New estimates on the size of $(α, 2 α)$ -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 $(α, 2 α)$ -Fursterberg sets. This provides a quantitative impr…

math.CA2021★ 4 cited

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…

math.CO2020

On rich lenses in planar arrangements of circles and related problems

Esther Ezra, Orit E. Raz, Micha Sharir +1

We show that the maximum number of pairwise non-overlapping $k$ -rich lenses (lenses formed by at least $k$ circles) in an arrangement of $n$ circles in the plane is $O\left(\frac{n…

math.CA2020★ 11 cited

A note on Fourier restriction and nested Polynomial Wolff axioms

Jonathan Hickman, Joshua Zahl

This note records an asymptotic improvement on the known $L^{p}$ range for the Fourier restriction conjecture in high dimensions. This is obtained by combining Guth's polynomial part…

math.MG2020★ 1 cited

Dimension-expanding polynomials and the discretized Elekes-Rónyai theorem

Orit E. Raz, Joshua Zahl

We characterize when bivariate real analytic functions are "dimension expanding" when applied to a Cartesian product. If $P$ is a bivariate real analytic function that is not local…

math.CO2020★ 9 cited

Distinct distances in the complex plane

Adam Sheffer, Joshua Zahl

We prove that if $P$ is a set of $n$ points in $C^{2}$ , then either the points in $P$ determine $Ω (n^{1 - ε})$ complex distances, or $P$ is contained in a line with slope $\pm…

math.CO2020★ 2 cited

Sphere tangencies, line incidences, and Lie's line-sphere correspondence

Joshua Zahl

Two spheres with centers $p$ and $q$ and signed radii $r$ and $s$ are said to be in contact if $∣ p - q ∣^{2} = (r - s)^{2}$ . Using Lie's line-sphere correspondence, we show that if $F$ is a…

math.CA2019★ 15 cited

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…

math.CA2019★ 10 cited

Large Sets Avoiding Rough Patterns

Jacob Denson, Malabika Pramanik, Joshua Zahl

The pattern avoidance problem seeks to construct a set $X \subset R^{d}$ with large dimension that avoids a prescribed pattern. Examples of such patterns include three-term a…

cs.CG2019★ 3 cited

Constructive Polynomial Partitioning for Algebraic Curves in $R^{3}$ with Applications

Boris Aronov, Esther Ezra, Joshua Zahl

In 2015, Guth proved that for any set of $k$ -dimensional bounded complexity varieties in $R^{d}$ and for any positive integer $D$ , there exists a polynomial of degree at mos…

On planar Nikodym-type sets

A streamlined proof of the Kakeya set conjecture in R3

On the number of pairwise touching cylinders in Rd

A Survey of the Kakeya conjecture, 2000-2025

A bound for plany Kakeya sets in Fq4​ using the planebrush method

Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions

Improved Lp bounds for the strong spherical maximal operator

The Assouad dimension of Kakeya sets in R3

On Maximal Functions Associated to Families of Curves in the Plane

Improved Elekes-Szabó type estimates using proximity

Kakeya sets from lines in SL2​

Sticky Kakeya sets and the sticky Kakeya conjecture

Unions of lines in Rn

A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in R3

New estimates on the size of (α,2α)-Furstenberg sets

On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-Rónyai theorem

On rich lenses in planar arrangements of circles and related problems

A note on Fourier restriction and nested Polynomial Wolff axioms

Dimension-expanding polynomials and the discretized Elekes-Rónyai theorem

Distinct distances in the complex plane

Sphere tangencies, line incidences, and Lie's line-sphere correspondence

New Kakeya estimates using Gromov's algebraic lemma

Large Sets Avoiding Rough Patterns

Constructive Polynomial Partitioning for Algebraic Curves in R3 with Applications

A streamlined proof of the Kakeya set conjecture in $R^{3}$

On the number of pairwise touching cylinders in $R^{d}$

A bound for plany Kakeya sets in $F_{q}^{4}$ using the planebrush method

Improved $L^{p}$ bounds for the strong spherical maximal operator

The Assouad dimension of Kakeya sets in $R^{3}$

Kakeya sets from lines in $S L_{2}$

Unions of lines in $R^{n}$

A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in $R^{3}$

New estimates on the size of $(α, 2 α)$ -Furstenberg sets

Constructive Polynomial Partitioning for Algebraic Curves in $R^{3}$ with Applications