Here’s one way to think about the Minkowski dimension: Take your set and cover it with tiny balls that each have a diameter of one-millionth of your preferred unit. If your set is a line segment of length 1, you’ll need at least 1 million balls to cover it. If your set is a square of area 1, you’ll need many, many more: a million squared, or a trillion. For a sphere of volume 1, it’s about 1 million cubed (a quintillion), and so on. The Minkowski dimension is the value of this exponent. It measures the rate at which the number of balls you need to cover your set grows as the diameter of each ball gets smaller. A line segment has dimension 1, a square has dimension 2, and a cube has dimension 3.
These dimensions are familiar. But using Minkowski’s definition, it becomes possible to construct a set that has a dimension of, say, 2.7. Though such a set doesn’t fill up three-dimensional space, it’s in some sense “bigger” than a two-dimensional surface.
When you cover a set with balls of a given diameter, you’re approximating the volume of the fattened-up version of the set. The more slowly the volume of the set decreases with the size of your needle, the more balls you need to cover it. You can therefore rewrite Davies’ result — which states that the area of a Kakeya set in the plane decreases slowly — to show that the set must have a Minkowski dimension of 2. The Kakeya conjecture generalizes this claim to higher dimensions: A Kakeya set must always have the same dimension as the space it inhabits.
That simple statement has been surprisingly difficult to prove.
A Tower of Conjectures
Until Fefferman made a startling discovery in 1971, the conjecture was viewed as a curiosity.
He was working on an entirely different problem at the time. He wanted to understand the Fourier transform, a powerful tool that allows mathematicians to study functions by writing them as sums of sine waves. Think of a musical note, which is made up of lots of overlapping frequencies. (That’s why a middle C on a piano sounds different from a middle C on a violin.) The Fourier transform allows mathematicians to calculate the constituent frequencies of a particular note. The same principle works for sounds as complicated as human speech.
Mathematicians also want to know whether they can rebuild the original function if they’re given just some of its infinitely many constituent frequencies. They have a good understanding of how to do this in one dimension. But in higher dimensions, they can make different choices about which frequencies to use and which to ignore. Fefferman proved, to his colleagues’ surprise, that you might fail to rebuild your function when relying on a particularly well-known way of choosing frequencies.
His proof hinged on constructing a function by modifying Besicovitch’s Kakeya set. This later inspired mathematicians to develop a hierarchy of conjectures about the higher-dimensional behavior of the Fourier transform. Today, the hierarchy even includes conjectures about the behavior of important partial differential equations in physics, like the Schrödinger equation. Each conjecture in the hierarchy automatically implies the one below it.
The Kakeya conjecture lies at the very base of this tower. If it is false, then so are the statements higher in the hierarchy. On the other hand, proving it true wouldn’t immediately imply the truth of the conjectures located above it, but it might provide tools and insights for attacking them.
“The amazing thing about the Kakeya conjecture is that it’s not just a fun problem; it’s a real theoretical bottleneck,” Hickman said. “We don’t understand a lot of these phenomena in partial differential equations and Fourier analysis because we don’t understand these Kakeya sets.”
Hatching a Plan
Fefferman’s proof — along with subsequently discovered connections to number theory, combinatorics and other areas — revived interest in the Kakeya problem among top mathematicians.
In 1995, Thomas Wolff proved that the Minkowski dimension of a Kakeya set in 3D space has to be at least 2.5. That lower bound turned out to be difficult to increase. Then, in 1999, the mathematicians Nets Katz, Izabella Łaba and Terence Tao managed to beat it. Their new bound: 2.500000001. Despite how small the improvement was, it overcame a massive theoretical barrier. Their paper was published in the Annals of Mathematics, the field’s most prestigious journal.
Katz and Tao later hoped to apply some of the ideas from that work to attack the 3D Kakeya conjecture in a different way. They hypothesized that any counterexample must have three particular properties, and that the coexistence of those properties must lead to a contradiction. If they could prove this, it would mean that the Kakeya conjecture was true in three dimensions.
They couldn’t go all the way, but they did make some progress. In particular, they (along with other mathematicians) showed that any counterexample must have two of the three properties. It must be “plany,” which means that whenever line segments intersect at a point, those segments also lie nearly in the same plane. It must also be “grainy,” which requires that the planes of nearby points of intersection be similarly oriented.
That left the third property. In a “sticky” set, line segments that point in nearly the same direction also have to be located close to each other in space. Katz and Tao couldn’t prove that all counterexamples must be sticky. But intuitively, a sticky set seems like the best way to force a lot of overlap among the line segments, thereby making the set as small as possible — precisely what you need to create a counterexample. If someone could show that a sticky Kakeya set had a Minkowski dimension of less than 3, it would disprove the 3D Kakeya conjecture. “It sounds like ‘sticky’ would be the most worrisome case,” said Larry Guth of the Massachusetts Institute of Technology.
It’s no longer a worry.
The Sticking Point
In 2014 — more than a decade after Katz and Tao attempted to prove the Kakeya conjecture — Tao posted an outline of their approach on his blog, giving other mathematicians the chance to try it out for themselves.
In 2021, Hong Wang, a mathematician at New York University, and Joshua Zahl of the University of British Columbia decided to pick up where Tao and Katz had left off.
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