# A Question About a Rotating Line Helps Reveal What Makes Real Numbers Special

## Introduction

The Kakeya conjecture sounds like a brain teaser. Place a needle flat on a table. How much area do you need in order to be able to turn it so that it points in all possible directions?

The most obvious possible answer is a circle whose diameter is the length of the needle. But this is demonstrably wrong. And over the last century, the effort to understand the ways in which it’s wrong has revealed that what seems like a fun little question is in fact a deeply provocative mathematical problem about the nature of the real numbers themselves — those infinite ticks of the number line that serve as the coordinates in space where the problem was first posed.

This has become clear with several proofs that have recently made some of the most notable progress in years on the Kakeya conjecture. The results transport the original question away from the real numbers, where mathematicians have been stymied, and into geometric and arithmetic worlds where lines are defined by alternative number systems that are in some ways easier to work with.

The inventiveness has animated mathematicians with a new sense of purpose.

“The Kakeya conjecture feels so hard, but it’s also plausible there will be a solution in a few years,” said Larry Guth, a mathematician at the Massachusetts Institute of Technology who has worked on the problem for more than 15 years. “It seems more hopeful than any other approach I’ve seen.”

**A Tight Squeeze**

The modern versions of the Kakeya conjecture are a few steps removed from the original statement of the problem, made in 1917 by Sōichi Kakeya. He was curious about the smallest area required in the two-dimensional plane to turn a one-dimensional line of a given length so that it eventually points in all directions.

A disk with a diameter equal to the length of the line suffices — just rotate the line like a dial. But smaller shapes can also work. For example, take an equilateral triangle with a height equal to the length of the line. By performing a series of what are essentially three-point turns, you can shift the line — which has zero area, because it’s one-dimensional — around the triangle and achieve the desired sweep. A set of points that enables this thorough pointing is called a Kakeya set.

Kakeya wanted to know the smallest possible area of a Kakeya set. In 1919, Abram Besicovitch provided the surprising answer: There is no limit to how small it can be. He demonstrated that it’s possible to construct Kakeya sets that take the equilateral-triangle design to its extreme. Instead of the three spikes of the triangle, you end up with a profusion of spikes within spikes emanating in all directions.

“In the limit it is a weird hedgehog-looking thing,” said Zeev Dvir, a professor at Princeton University and an author of one of the new proofs. The result is an intricate fractal arrangement with an area that can be made arbitrarily small — which is tantamount to having no area at all.

David Kelly Crow

Besicovitch’s construction took the air out of Kakeya’s question just two years after he asked it. But decades later, mathematicians devised a revised version of the problem that would prove far more bedeviling.

**Widespread Emptiness**

Besicovitch proved that Kakeya sets can have vanishing area, but there are ways other than area to describe the size of a shape. The sets that Besicovitch devised still contain points, and in the 1970s a revamped question arose about how efficiently those points are arranged.

This question, called the Kakeya conjecture (as distinct from the original Kakeya problem), predicts that if you have, say, little squares of fabric, and you’re trying to position them over a Kakeya set so that the squares cover the set completely, in some very precise sense you’ll need a lot of squares to complete the covering.

The extent to which points in a set are arranged in a way that makes them easier or harder to cover is captured in two closely related metrics called Hausdorff dimension and Minkowski dimension. These notions of dimension provide mathematicians with another rigorous framework in which to explore the Kakeya sets — a way of continuing to investigate them after Besicovitch proved that measuring area alone was insufficient for understanding their essential properties. The Kakeya conjecture predicts that both the Hausdorff and the Minkowski dimensions of a Kakeya set must be as large as possible. And while the exact definitions of those two measures of dimension are technical, the intuition behind the conjecture is fairly plain: To have lines going everywhere, you need a lot of something.

“You have one line in every direction, and imagine you’re trying to squish them all into something. How could it get compressed?” said Guth.

**Real Problems**

The Kakeya conjecture takes place in Euclidean space, where points are defined by real numbers — numbers that can have an infinitely long decimal, like 19.1777… or pi. Over time it’s become clear that these real-value coordinates are a big part of why the Kakeya conjecture is so hard to solve.

Courtesy of the Graduate School of Mathematical Sciences, University of Tokyo

Exactly what it is about the real numbers that creates such an obstruction is not entirely clear, but some features stand out. First, the real numbers are continuous, which implies that you can’t look at them over any discrete interval without losing the ability to do arithmetic. (If you restrict yourself to an interval between 1 and 2, for example, you lose addition, because the sum of two numbers within that interval will lie outside it.) The real numbers are also uncountably infinite, which means that no matter how much you zoom in on them, you see the same thing at every scale.

“In the real numbers, things can be very close to zero without actually being zero. Somehow that is the technical crux,” said Joshua Zahl of the University of British Columbia.

The difficulty of the real numbers has motivated mathematicians to consider versions of the Kakeya conjecture that are set in smaller number systems. These might only have the whole number values 1 through 5, for example. And while these number systems don’t look much like the real numbers, they carry many of the same basic arithmetic properties — they allow addition, subtraction, multiplication and division.

They are also rich enough to support the techniques from linear algebra for defining lines, and once you have lines you can ask a slightly modified version of the Kakeya conjecture: What is the minimum size of a set of points in one of these number systems such that you can construct a line in every direction? Thomas Wolff asked a question like this in 1996. Since then, mathematicians have approached it as a scaffold that could bring them closer to answering the Kakeya conjecture itself.

“The idea is [this] problem is presumably easier, and maybe you should try developing techniques for solving that to get ideas for dealing with the actual Euclidean case,” said Manik Dhar of Princeton, an author of two recent papers on the Kakeya conjecture.

**Pick a Number**

To define one of these small number systems you first pick a number. Maybe you choose 9, in which case your number system contains the whole numbers 1 through 9. Or maybe you choose 17, 25 or 83.

Your choice matters. In particular, whether this number (called the modulus) is prime or not prime, and what way it’s not prime, has a big effect on both the behavior of the number system and the methods that might apply to the Kakeya conjecture.

David Kelly Crow

In 2008, Dvir solved the Kakeya conjecture for finite number systems in which the modulus is a prime number, which is the particular case Wolff had in mind in 1996. These number systems, called finite fields, are especially powerful and are used throughout mathematics to attack hard problems.

Dvir proved that over finite fields, a Kakeya set necessarily has the largest possible dimension (where dimension is redefined in a way that makes sense in a finite setting). His proof, which was just two pages long, leaned heavily on the fact that when the modulus is prime, any set within the finite number system serves as the solutions (or roots) to a polynomial equation — meaning the set can be described by an equation in a way that real-number Kakeya sets cannot be.

Dvir’s proof represented the first major progress on the Kakeya conjecture and made mathematicians momentarily hopeful that further advances toward the Euclidean Kakeya conjecture were in store.

None turned up. “People were very excited and we all tried hard, and it didn’t work,” said Guth.

Then, more than a decade later, Dvir came back.

**Products of Primes**

In November 2020, Dvir and Dhar, his graduate student, solved the Kakeya conjecture for finite number systems in which the modulus is any number that is the product of distinct primes, like 15 (which is 3 × 5). These number systems required Dhar and Dvir to move beyond the polynomial method. Instead, they converted the problem into a question about tables of numbers called matrices.

In these matrices, columns represent points and rows represent directions. If there’s a line at a particular point, going in a particular direction, write a 1 in the corresponding spot in the matrix. (Otherwise enter 0.) In this way, the matrix encodes the properties of a set of lines. Now you can calculate properties of that matrix to determine properties of the set. In particular, the matrix’s “rank” relates directly to the size of the set of lines.

Ana Arsovska

Dhar and Dvir proved that the rank of these matrices is high, which means the set of lines is large, which means the Kakeya conjecture is true for these particular number systems — any set of points containing lines in all directions needs to be big.

Less than a year after Dhar and Dvir’s result, Bodan Arsovski extended it. In August 2021 he proved the Kakeya conjecture for finite number systems in which the modulus is a prime number raised to a power, such as 9 (which is 3^{2}). This implies the conjecture for a number system called the *p*-adics, which is an infinite number system and more like the real numbers in that way. Following Arsovski’s paper, mathematicians threw themselves into determining whether his methods could be modified to apply to the real numbers themselves.

After a few months of fruitless effort, it became apparent that for now, at least, they can’t be.

“There are small differences in how the field of real numbers and *p*-adic fields behave that make the analogy kind of break,” said Alejo Salvatore, a doctoral student at the University of Wisconsin, Madison.

Since Arsovski’s work there have been two more plot twists. Last October Dhar proved the Kakeya conjecture true for finite number systems with any modulus. Then in February Salvatore confirmed the conjecture for more exotic number systems, called local fields of positive characteristic, in which a finite field is augmented with a variable.

There are different ways to think about this flurry of results. One is to hope that the momentum continues: Now that mathematicians have proved the conjecture true for one number system after another, perhaps the real numbers are next. But another is to step back and ask: Why haven’t mathematicians been able to confirm the Kakeya conjecture for the real numbers, given that they’ve now been able to confirm it in so many other settings?

At least one mathematician thinks the explanation might be the most obvious one of all.

“I’m no longer confident that the Kakeya conjecture is true,” said Guth.