A streamlined proof of the Kakeya set conjecture in $\mathbb{R}^3$
We present a streamlined and simplified proof of the Kakeya set conjecture in $\mathbb{R}^3$.
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arXiv subjects
Joshua Zahl
Publications and source records attributed to Joshua Zahl.
A Survey of the Kakeya conjecture, 2000-2025
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.
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On Maximal Functions Associated to Families of Curves in the Plane
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 function, the circular maximal functions of Wolff and Bourgain, and their multi-parameter analogues. We propose a framework that allows for a unified study of such maximal functions, and prove sharp $L^p\to L^p$ operator bounds in this setting. A key ingredient is an estimate from discretized incidence geometry that controls the number of higher order approximate tangencies spanned by a collection of plane curves. We discuss applications to the Fässler-Orponen restricted projection problem, and the dimension of Furstenberg-type sets associated to families of curves.
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A Kakeya maximal function estimate in four dimensions using planebrushes
We obtain an improved Kakeya maximal function estimate and improved Kakeya Hausdorff dimension estimate in $\mathbb{R}^4$ using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff's hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates of Guth-Zahl can be used to bound the size of a Besicovitch set. In particular, we establish a maximal function estimate in $\mathbb{R}^4$ at dimension 3.049, and we prove that every Besicovitch set in $\mathbb{R}^4$ must have Hausdorff dimension at least 3.059.
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A bound for plany Kakeya sets in $\mathbb{F}_q^4$ using the planebrush method
Katz and Zahl used a planebrush argument to prove that Kakeya sets in $\mathbb{R}^4$ have Hausdorff dimension at least 3.059. In the special case when the Kakeya set is plany, their argument gives a better lower bound of 10/3. We give a nontechnical exposition of the Katz-Zahl argument for plany Kakeya sets in the finite field setting.
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Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions
We study sets of $δ$ tubes in $\mathbb{R}^3$, 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 set must have almost maximal volume. As a consequence, we prove that every Kakeya set in $\mathbb{R}^3$ has Minkowski and Hausdorff dimension 3.
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Improved $L^p$ bounds for the strong spherical maximal operator
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 operator is bounded on $L^p$ for $p > 2$ in all dimensions $n \geq 3$. This matches the conjectured sharp range $p>(n+1)/(n-1)$ when $n=3$. For $n=2$ the analogous estimate was recently proved by Chen, Guo and Yang. Our result builds upon and improves an earlier bound of Lee, Lee and Oh. The main novelty is an estimate in discretised incidence geometry that bounds the volume of the intersection of thin neighbourhoods of axis-parallel ellipsoids. This estimate is then interpolated with the Fourier analytic $L^p$-Sobolev estimates of Lee, Lee and Oh.
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A Furstenberg-type problem for circles, and a Kaufman-type restricted projection theorem in $\mathbb{R}^3$
We resolve a conjecture of Fässler and Orponen on the dimension of exceptional projections to one-dimensional subspaces indexed by a space curve in $\mathbb{R}^3$. We do this by obtaining sharp $L^p$ bounds for a variant of the Wolff circular maximal function over fractal sets for a class of $C^2$ curves related to Sogge's cinematic curvature condition. A key new tool is the use of lens cutting techniques from discrete geometry.
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The Assouad dimension of Kakeya sets in $\mathbb{R}^3$
This paper studies the structure of Kakeya sets in $\mathbb{R}^3$. We show that for every Kakeya set $K\subset\mathbb{R}^3$, there exist well-separated scales $0<\delta<\rho\leq 1$ so that the $\delta$ neighborhood of $K$ is almost as large as the $\rho$ neighborhood of $K$. As a consequence, every Kakeya set in $\mathbb{R}^3$ has Assouad dimension 3 and every Ahlfors-David regular Kakeya set in $\mathbb{R}^3$ has Hausdorff dimension 3. We also show that every Kakeya set in $\mathbb{R}^3$ that has "stably equal" Hausdorff and packing dimension (this is a new notion, which is introduced to avoid certain obvious obstructions) must have Hausdorff dimension 3. The above results follow from certain multi-scale structure theorems for arrangements of tubes and rectangular prisms in three dimensions, and a mild generalization of the sticky Kakeya theorem previously proved by the authors.
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On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-Rónyai theorem
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 theorems, in the spirit of Kaufman, Bourgain, and Shmerkin. (3) The parallelizability of planar $d$-webs. (4) The Elekes-Rónyai theorem on expanding polynomials. Given a Borel set $A$ in the plane, we study the set of exceptional vantage points, for which the pinned distance $Δ_p(A)$ has small dimension, that is, close to $(\dim A)/2$. We show that if this set has positive dimension, then it must have very special structure. This result follows from a more general single-scale nonlinear projection theorem, which says that if $ϕ_1,ϕ_2,ϕ_3$ are three smooth functions whose associated 3-web has non-vanishing Blaschke curvature, and if $A$ is a $(δ,α)_2$-set in the sense of Katz and Tao, then at least one of the images $ϕ_i(A)$ must have measure much larger than $|A|^{1/2}$, where $|A|$ stands for the measure of $A$. We prove analogous results for $d$ smooth functions $ϕ_1,\ldots,ϕ_d$, whose associated $d$-web is not parallelizable. We use similar tools to characterize when bivariate real analytic functions are "dimension expanding" when applied to a Cartesian product: if $P$ is a bivariate real analytic function, then $P$ is either locally of the form $h(a(x) + b(y))$, or $P(A,B)$ has dimension at least $α+c$ whenever $A$ and $B$ are Borel sets with Hausdorff dimension $α$. Again, this follows from a single-scale estimate, which is an analogue of the Elekes-Rónyai theorem in the setting of the Katz-Tao discretized ring conjecture.
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Kakeya sets from lines in $SL_2$
We prove that every Kakeya set in $\mathbb{R}^3$ formed from lines of the form $(a,b,0) + \operatorname{span}(c,d,1)$ with $ad-bc=1$ must have Hausdorff dimension $3$; Kakeya sets of this type are called $SL_2$ Kakeya sets. This result was also recently proved by Fässler and Orponen using different techniques. Our method combines induction on scales with a special structural property of $SL_2$ Kakeya sets, which says that locally such sets look like the pre-image of an arrangement of plane curves above a special type of map from $\mathbb{R}^3$ to $\mathbb{R}^2$, called a twisting projection. This reduces the study of $SL_2$ Kakeya sets to a Kakeya-type problem for plane curves; the latter is analyzed using a variant of Wolff's circular maximal function.
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Improved Elekes-Szabó type estimates using proximity
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 particular, if $A,B,C$ are sets of $N$ real numbers and $f$ is a trivariate polynomial, then either $f$ has a special form that encodes additive group structure (for example $f(x,y,x) = x + y - z$), or $A \times B\times C \cap\{f=0\}$ has cardinality $O(N^{12/7})$. This is an improvement over the previously bound $O(N^{11/6})$. We also prove an asymmetric version of our main result, which yields an Elekes-Ronyai type expanding polynomial estimate with exponent $3/2$. This has applications to questions in combinatorial geometry related to the Erdős distinct distances problem. Like previous approaches to the problem, we rephrase the question as a $L^2$ estimate, which can be analyzed by counting additive quadruples. The latter problem can be recast as an incidence problem involving points and curves in the plane. The new idea in our proof is that we use the order structure of the reals to restrict attention to a smaller collection of proximate additive quadruples.
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New estimates on the size of $(α,2α)$-Furstenberg sets
We use recent advances on the discretized sum-product problem to obtain new bounds on the Hausdorff dimension of planar $(α,2α)$-Fursterberg sets. This provides a quantitative improvement to the $2α+ε$ bound of Héra-Shmerkin-Yavicoli. In particular, we show that every $1/2$-Furstenberg set has dimension at least $1 + 1/4536$.
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Sticky Kakeya sets and the sticky Kakeya conjecture
A Kakeya set is a compact subset of $\mathbb{R}^n$ that contains a unit line segment pointing in every direction. The Kakeya conjecture asserts that such sets must have Hausdorff and Minkowski dimension $n$. There is a special class of Kakeya sets, called sticky Kakeya sets. Sticky Kakeya sets exhibit an approximate multi-scale self-similarity, and sets of this type played an important role in Katz, {\L}aba, and Tao's groundbreaking 1999 work on the Kakeya problem. We propose a special case of the Kakeya conjecture, which asserts that sticky Kakeya sets must have Hausdorff and Minkowski dimension $n$. We prove this conjecture in three dimensions.
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Unions of lines in $\mathbb{R}^n$
We prove a conjecture of D. Oberlin on the dimension of unions of lines in $\mathbb{R}^n$. If $d \geq 1$ is an integer, $0 \leq β\leq 1$, and $L$ is a set of lines in $\mathbb{R}^n$ with Hausdorff dimension at least $2(d-1) + β$, then the union of the lines in $L$ has Hausdorff dimension at least $d + β$. Our proof combines a refined version of the multilinear Kakeya theorem by Carbery and Valdimarsson with the multilinear to linear argument of Bourgain and Guth.
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Breaking the 3/2 barrier for unit distances in three dimensions
We prove that every set of $n$ points in $\mathbb{R}^3$ spans $O(n^{295/197+ε})$ unit distances. This is an improvement over the previous bound of $O(n^{3/2})$. A key ingredient in the proof is a new result for cutting circles in $\mathbb{R}^3$ into pseudo-segments.
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Dimension-expanding polynomials and the discretized Elekes-Rónyai theorem
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 locally of the form $P(x,y) = h(a(x) + b(y))$, then whenever $A$ and $B$ are Borel subsets of $\mathbb{R}$ with Hausdorff dimension $0<α<1$, we have that $P(A,B)$ has Hausdorff dimension at least $α+ ε$ for some $ε(α)>0$ that is independent of $P$. The result is sharp, in the sense that no estimate of this form can hold if $P(x,y) = h(a(x) + b(y))$. We also prove a more technical single-scale version of this result, which is an analogue of the Elekes-Rónyai theorem in the setting of the Katz-Tao discretized ring conjecture. As an application, we show that a discretized non-concentrated set cannot have small nonlinear projection under three distinct analytic projection functions, provided that the corresponding 3-web has non-vanishing Blaschke curvature.
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A discretized Severi-type theorem with applications to harmonic analysis
In 1901, Severi proved that if $Z$ is an irreducible hypersurface in $\mathbb{P}^4(\mathbb{C})$ that contains a three dimensional set of lines, then $Z$ is either a quadratic hypersurface or a scroll of planes. We prove a discretized version of this result for hypersurfaces in $\mathbb{R}^4$. As an application, we prove that at most $δ^{-2-\varepsilon}$ direction-separated $δ$-tubes can be contained in the $δ$-neighborhood of a low-degree hypersurface in $\mathbb{R}^4$. This result leads to improved bounds on the restriction and Kakeya problems in $\mathbb{R}^4$. Combined with previous work of Guth and the author, this result implies a Kakeya maximal function estimate at dimension $3+1/28$, which is an improvement over the previous bound of $3$ due to Wolff. As a consequence, we prove that every Besicovitch set in $\mathbb{R}^4$ must have Hausdorff dimension at least $3+1/28$. Recently, Demeter showed that any improvement over Wolff's bound for the Kakeya maximal function yields new bounds on the restriction problem for the paraboloid in $\mathbb{R}^4$.
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