‘Once in a Century’ Proof Settles Math’s Kakeya Conjecture
Introduction
Consider a pencil lying on your desk. Try to spin it around so that it points once in every direction, but make sure it sweeps over as little of the desk’s surface as possible. You might twirl the pencil about its middle, tracing out a circle. But if you slide it in clever ways, you can do much better.
“It’s just a problem about how straight lines can intersect one another,” said Jonathan Hickman, a mathematician at the University of Edinburgh. “But there’s such an incredible richness encoded in it — an incredible array of connections to other problems.”
For five decades, mathematicians have sought the best possible solution to the three-dimensional version of this challenge: Hold a pencil in midair, then point it in every direction while minimizing the volume of space it moves through. This straightforward problem has eluded some of the greatest living mathematicians, and it lurks beneath a host of open problems.
Now, the hunt for a solution appears to be over. In a paper recently posted on the scientific preprint site arxiv.org, Hong Wang of New York University’s Courant Institute and Joshua Zahl of the University of British Columbia have proved the three-dimensional Kakeya conjecture — they’ve established an absolute limit to how small such a pattern of movements can be.
“This thing doesn’t need hyping up,” said Nets Katz, a mathematician at Rice University. “It’s a once-in-a-century kind of result.”
A Thickening Plot
In 1917, Sōichi Kakeya posed the problem, but with an infinitely thin pencil. He found a way of sliding the pencil that covered less area than the instinctual circular motion.
Kakeya wondered how small an area the pencil could possibly sweep. Two years later, the Russian mathematician Abram Besicovitch found the answer: a complicated set of narrow turns that, counterintuitively, covers no space at all.
That more or less settled the question until 1971, when Robert Fefferman was studying something apparently unrelated to twirling lines: the Fourier transform, a foundational mathematical tool that lets you reimagine any mathematical function as a combination of waves. In Fefferman’s work, a tweaked version of Kakeya’s problem kept coming up. In this case, the pencil has a thickness and twirls in three dimensions. Here, Kakeya’s question becomes the following: As you change the width of the pencil, how does it affect the volume of space that it traces out?
Mathematicians prefer to picture this problem in a slightly different (but equivalent) way. Instead of moving a pencil around in space, imagine every position in the pencil’s trajectory, all at once. What you get is a configuration of ghostly, overlapping tubes pointing everywhere, called a Kakeya set. You can slide the tubes around, but you can’t rotate them. Your goal is to form a configuration with the most overlap.
Hong Wang, a mathematician at the Courant Institute at New York University, said the proof will open up new vistas in mathematics. “It needed to be done,” she said.
Rickinasia/Wikimedia Commons
Even the Kakeya set that overlaps the most has to take up some space, Fefferman found. That minimum volume depends on how thick the tubes are. Mathematicians quantify the relationship between the tubes’ thickness and the volume of the set using a number called the Minkowski dimension. The smaller the Minkowski dimension, the more you can reduce the set’s volume by thinning the tubes slightly.
The three-dimensional Kakeya conjecture says that a set’s Minkowski dimension must be three. This constitutes a very weak relationship — if you halve the tubes’ thickness, for instance, you will only remove a sliver of the volume at most.
Yet even that mild constraint turned out to be nearly impossible to prove.
Baby Steps
In 2022, five decades after the modern Kakeya conjecture was formulated, Wang and Zahl took a significant step forward. Following a program that Katz and Terence Tao had laid out back in 2014, they examined a pesky class of Kakeya sets. Their proof showed that every set in that particular class had a dimension of three. (The proof applies to both the Minkowski dimension and a closely related concept called the Hausdorff dimension.) With that annoying group set aside, they now had to show that the dimension was three for all the other Kakeya sets.
Their approach was to go step-by-step. They would first examine a narrow range of Minkowski dimensions — say, 2.5 to 2.6 — and try to show that no Kakeya set could be in that range. If they could prove this for every interval up to three, they’d prove the Kakeya conjecture.
Fortunately, Wang and Zahl didn’t have to start from zero. Tom Wolff proved in 1995 that no three-dimensional Kakeya set has a Hausdorff or Minkowski dimension below 2.5. But they needed a way to prove that a dimension between 2.5 and, say, 2.500001, was also impossible. Then they could repeat that argument to get a bound of 2.500002, and so on. Each time, they would essentially be showing that no Kakeya sets exist within that tiny increment.
In practice, they didn’t actually have to tediously prove each of these millions of increments one by one. They just needed to prove the first increment, so long as they could show that one bound implies the next, slightly larger one. Then they had to show that their argument worked no matter where they began. That would be enough to show that the bound can be walked up all the way to three.
But unlike in 2022, when they used Katz and Tao’s strategy, they had no road map to follow. They turned to a special property called graininess.
In 2014, Larry Guth, a mathematician at the Massachusetts Institute of Technology, had proved that any counterexample to the Kakeya conjecture needed to be “grainy.” In a grainy set, there are many small 3D sections where lots of tubes overlap. Each of these “grains” is about one tube thick and a few times wider, but not nearly as long, with many tubes passing through it lengthwise.
Wang and Zahl realized they could eschew the tubes entirely and deal with these simpler grains. They found that it was easier to enumerate and calculate the various ways the grains could overlap.
Joshua Zahl, a mathematician at the University of British Columbia, co-authored the new proof.
Paul Joseph
And even in cases where the grains all conspired to provide maximum overlap, they found, the number of grains intersecting any given point couldn’t be too big. Starting from the 2.5 bound, they were able to prove that the grains couldn’t overlap enough to result in a dimension slightly above that bound either. Then, starting from the higher bound, they showed that the same computational steps could be applied to nudge the bound even higher. And so on.
“It’s like perfecting a perpetual-motion machine. It’s magical,” Tao said. “They’re getting more at the output than the input.” Their machine took them all the way to a Minkowski (and Hausdorff) dimension of three, proving the three-dimensional Kakeya conjecture.
Tower of Dreams
The conjecture’s resolution is a seismic shift for the field of harmonic analysis, which studies the details of the Fourier transform.
A tower of three monumental conjectures in harmonic analysis rests atop the Kakeya conjecture. Each story in the tower needs to be sturdy for the stories above it to stand a chance themselves. If the Kakeya conjecture had been proved false — if Wang and Zahl had found a counterexample — the entire tower would have come tumbling down.
But now that they’ve proved it, mathematicians might be able to work their way up the tower, using Kakeya to build up proofs of these successively more ambitious conjectures. “All these problems that [mathematicians] dreamed about someday solving, they all look approachable now,” Guth said.
That process has already begun. Wang recently co-authored a separate paper reducing the next conjecture in the tower to a stronger version of the Kakeya conjecture, a step toward bridging the two levels.
It’s also a dimensional leap for this entire area of math that’s been somewhat stuck in 2D. “People understood what’s going on [in Kakeya-adjacent problems] really well in two dimensions, but we lacked the tools to study higher dimensions,” Wang said. “So I feel like this was necessary. It needed to be done.”
The four-dimensional Kakeya conjecture remains open, with a tower of four-dimensional conjectures above it as well. New difficulties will arise, Guth said, but he thinks that the jump from two dimensions to three was the hardest, and that Wang and Zahl’s proof can likely be adapted to that tower, and beyond.
“When I got excited about the Kakeya problem as a younger mathematician, it just felt so simple and geometrical that it was surprising to me that it was hard,” Guth said. Years later, Wang, his doctoral student, was motivated by the same deceptive simplicity.
“You have these concrete things you can visualize. It’s not as scary as other math theories,” Wang said. “I just wanted to understand why it’s hard.”
Now, thanks to Wang and Zahl’s efforts, that understanding is closer than ever. “I really think there’s a critical mass of ideas to really revolutionize the whole field coming from here,” Hickman said. “It’s a very, very exciting time.”
