Hong Wang and Joshua Zahl’s Solution for Kakeya’s Problem in Three Dimensions – Reflections and Links

Combinatorics and more 2025-04-05

As many of you likely heard by now, Hong Wang and Joshua Zahl proved the Kakeya’s Conjecture in three dimensions.

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

Wang and Zahl’s spectacular work was described in Terry Tao’s blog, in an article by Joseph Howlett in Quanta Magazine, and in many other places. Congratulations to Hong and Joshua!

Top: Joshua Zahl and Hong Wang lecturing on the result, alongside a figure from the paper. Bottom: Sōichi Kakeya, Abram Besikovich, Guido Fubini and Tom Wolff. (Click to enlarge.)

This post includes some reflections on Kakeya’s conjecture, along with some (mostly older) related information.

The Kakeya problem

A Kakeya set in $\mathbb R^d$ is a set $K$ that contains a unit interval in any direction.

The Kakeya Conjecture states that the dimension of a Kakeya set is at least $d$ . The conjecture applies to both Minkowski’s dimension and Hausdorf dimension. Roy Davies proved the conjecture in the plane in 1971.

For a comprehensive overview of the problem and its connections to harmonic analysis, combinatorial geometry, incidence theory, finite field analogs, and additive combinatorics, see Isabella Laba’s 2008 survey article From harmonic analysis to arithmetic combinatorics.

Overstretching Fubini’s theorem

What about the measure of a Kakeya set? It is known that a Kakeya set can have measure zero—a delicate fact that seems counterintuitive at first. Naïvely, Fubini’s theorem might suggest that the measure should be positive, or even at least one. Consider the planar case: “compute” the area by taking integral over all directions and in each direction choose the line whose intersection with the set have length at least one. Besikovich constructed a Kakeya set of measure zero. (See his 1928 lovely to read English paper on the topic, as well as his 1963 American Mathematical Monthly paper).

A word about Fubini’s theorem: this famous result essentially states that when computing the volume of a set (or evaluating integrals in general), the order of integration does not matter—though many caveats apply. A similar naïve application of Fubini’s theorem arises in the slicing conjecture (mentioned in this post) and various other problems. (“Computing” the volume of a convex body in $d$ -dimensions by integrating the $(d-1)$ volumes of its sections in all direction would have given the constant 1 for the slicing conjecture; but this is incorrect.)

Fubini’s theorem is also related to the widely used combinatorial method of counting pairs in two different ways.

Minkowski and Hausdorff dimension and fractals.

The Minkowski’s dimension of a set $X$ is based on counting the number $N(\epsilon)$ of standard boxes with edge length $\epsilon$ required to cover $X$ . If $N(\epsilon )$ behaves like $\epsilon ^{-d}$ then $X$ has dimension $d$ . The Hausdorff $d$ -measure is an extension of Lebesgue measure based on covering the set with countable sets of diameter at most $\delta$ . The $d$ -measure is defined as the infimum of the sums of the $d$ -th power of the diameters of the parts. The Housdorff dimension is the infimum value of $d$ such that the Housdorff $d$ -measure is zero. Here is a post about fractals and their fractional dimensions. A recommended (freely available!) book is Bishop and Peres’s Fractals in Probability and Analysis. And, here is a videotaped lecture by Kenneth Falconer on intermediate dimensions capacities and projections.

Harmonic analysis connections

Over the past five decades, it was discovered that the Kakeya conjecture is related to a whole array of conjectures in harmonic analysis. Laba’s survey article discusses some of these connections and further connections to additive combinatorics. A somewhat stronger version of the Kakeya conjecture is the Kakeya maximal function conjecture (which is still open in three dimensions). Another related conjecture is the restriction problem (posed by Elias Stein). The conjectural answer of the restriction problem would imply the Kakeya conjecture. Here is a blog post by Terry Tao on the restriction problem and lecture notes also by Tao, and here is a related videotaped lecture by Larry Guth.

In a recent breakthrough, Hannah Cairo disproved two major conjectures: the Mizohata-Takeuchi conjecture, and an even stronger 1978 Stein’s conjecture. (h/t Tarry Tao on Mathstodon.xyz over Mastodon.)

The Kakeya problem and the restriction problem, Edinburgh 2002

The peak of my understanding of these problems came while attending a summer school in Edinburgh in 2002, where two lecture series were dedicated to the restriction and Kakeya problems, along with several talks on related topics. I recall that the type of inequalities involved seemed (vaguely) related to the hypercontractive inequalities that played a part in the analysis of Boolean functions—a topic I presented at that meeting. I also remember that the moment curve $(t,t^2,\dots, t^n)$ appeared in some work (perhaps by Bourgain) related to the restriction conjecture though I no longer recall the details.

I believe both Terry Tao and Tony Carbery gave lecture series on the topic. (Here are Tao’s lecture notes in dvi format.) Even more than the mathematics, I remember many younger and older mathematicians that I met in Edinburgh at that lovely meeting. It was also one of two mathematical occasions where I came with my wife, three children, my mother and my sister. (With the exception of Dvir’s 2008 theorem and its recent extension my post mainly describes development from before that meeting. .)

The connections to discrete and computational geometry and incidences.

In the last three decades beautiful and useful connections were found between the Kakeya conjecture and the study of incidences in discrete geometry (where a major starting problem was the Szemeredi-Trotter theorem; see this post). Here is a link to Zeev Dvir’s beautiful book Incidence Theorems and Their Applications, with a lot of related material.

The finite field analog of Kakeya’s problem.

Thomas Wolff raised a finite field analogue of the Kakeya problem and for some time, progress on both the real and finite field versions proceeded in parallel. The finite field conjecture was settled in 2008 by Zeev Dvir using the polynomial method. More recently Manik Dhar and Zeev Dvir proved a similar results for $(\mathbb X/N\mathbb Z)^m$ when $N$ is a square-free integer.

It is a very interesting while frustrating question to understand the connections between the finite field problem and the problem over the reals. Understanding the relationship between the finite field problem and its real counterpart remains a fascinating yet frustrating challenge. While extending the polynomial method to the real setting or applying the finite field case directly to Kakeya over the reals seems like natural approaches, neither is currently known, despite considerable effort in these directions. (I would certainly like to better understand the key ideas and obstacles in these two avenues.) A similar question arises for the connection between 3-AP free sets over $\mathbb Z/n\mathbb Z$ and over $(\mathbb X/p\mathbb Z)^m$ .

Another algebraic direction that captured my imagination is The p -adic Kakeya conjecture, by Bodan Arsovski. Having in analysis an interplay between questions over finite fields, local fields, number fields, and $\mathbb R, \mathbb C$ would be very nice.

AI: “Please, invent a new notion of dimension”

I asked ChatGPT about Minkowski and Hausdorff notions of dimension (and got pretty good answers), and then instructed it politely: “Please invent a new notion of dimension”. ChatGPT invented(?) a notion called “recursive covering dimension”, and wrote

“Recursive Covering Dimension is designed to measure the structural complexity of a set by taking into account how well it can be covered at multiple levels of recursion. Unlike the Hausdorff or box-counting dimension, which rely on a single scale-dependent covering, this dimension captures how efficiently a set can be described by nested coverings of progressively finer granularity.”

ChatGPT went on to describe the mathematical definition and properties of this notion; I have no idea how sensible, novel, correct or useful the definition is. (Try for yourself and see what notion of dimension AI will invent for you.) It looks that ChatGPT can already talk the talk but I am not sure if it really walks the walk.

Two remarks: 1) I had some disappointing experience with ChatGPT regarding the question: Compute the sum of fourth powers of irreducible representation of the symmetric group $S_n$ . ChatGPT insisted that the answer is $n!$ exactly like the sum of squares; again, try it for yourself.) 2) I also asked ChatGPT (always the free version): “Write your own blog post based on Hong Wang and Joshua Zahl’s proof of the Kakeya’s conjecture in three dimensions.” It wrote a nice post.