New Number Systems Point Geometry Problem Toward a Real Solution | Quanta Magazine

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.

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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 32). 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?

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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.

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