In a way, the integer distance problem was a victim of its own early successes. The hyperbola proof, with its ingenious simplicity, is emblematic of the philosophy espoused by Erdős, a highly influential mathematician who often spoke of “The Book” — an imagined volume of the most elegant proofs in mathematics. The culture of simplicity Erdős promoted has led to “tremendous results” in combinatorial geometry, Iosevich said. But it can also lead to blind spots — in this case, about the value of bringing in approaches from algebraic geometry.

“I don’t think you will find a result [in algebraic geometry] proven in the last 50 years that is not very technically involved and messy,” Iosevich said. “However, sometimes things need to be this way.”

In retrospect, the integer distance problem was waiting for mathematicians who were willing to consider more unruly curves than hyperbolas and then draw on recondite tools from algebraic geometry and number theory to tame them. “It required people with a sufficient breadth of knowledge and interest,” Iosevich said.

Most mathematicians, he said, are content to ply a few tools in one corner of mathematics for their entire career. But Greenfeld, Iliopoulou and Peluse are fearless explorers, Iosevich said. “They view mathematics as a coherent whole.”

Complexifying the Problem

In the summer of 2021, Greenfeld decided that it was time to take a stab at a problem from harmonic analysis she’d been mulling over since graduate school. Classical harmonic analysis, which forms the foundation for signal processing in the real world, is all about decomposing signals into sine waves of different frequencies and phases. This process works because it’s possible to make an infinite list of sine waves that, when combined, capture all the features of any signal, without any redundancy.

Often, though, researchers want to study something more complicated than a one-dimensional signal. For instance, they might want to decompose a signal on a disk in the plane. But the disk can only host a finite collection of compatible sine waves — too few to capture the behavior of all possible signals on the disk. The question then becomes: How big can this finite collection be?

In such a collection, the frequencies of the sines can be represented as points in the plane that seem averse to clustering in lines and circles: You’ll never find three points that are all close to the same line, or four that are all close to the same circle. Greenfeld hoped to use this aversion to prove that these sets of frequencies can contain only a few points.

At a 2021 meeting at the University of Bonn, Greenfeld attended a talk about the “determinant method,” a technique from number theory that can be used to estimate how many integer points of certain types can lie on curves. This tool, she realized, might be just what she needed. Greenfeld recruited Iliopoulou and Peluse, who were also at the meeting. “We started to learn this method together,” Greenfeld said.

But despite many efforts, they couldn’t seem to bend the determinant method to their purpose, and by the spring of 2023, they were feeling discouraged. Iosevich had invited Greenfeld and Peluse to drive to Rochester for a visit. “So we were thinking, ‘OK, we’ll go to Rochester, and talking to Alex will reinvigorate us,’” Peluse said. But as it turned out, they landed in Rochester already reinvigorated, thanks to a bracing discussion of integer distance sets on their unplanned detour along the Susquehanna River in Pennsylvania.

They arrived too late for a planned dinner with Iosevich, but they found him waiting in the hotel lobby with bags of takeout. He forgave their lateness — and was more than forgiving the next morning, when they told him about their plan to tackle integer distance sets. “He was so excited,” Peluse recalled. “Emotionally, this was a huge boost.”

As with the hyperbola approach, Greenfeld, Iliopoulou and Peluse tried to control the structure of integer distance sets by identifying families of curves the points must lie on. The hyperbola method starts to get too convoluted as soon as you have more than a few points, but Greenfeld, Iliopoulou and Peluse figured out how to consider many points at the same time by moving the entire configuration into a higher-dimensional space.

To see how this works, suppose you start with a “reference” point A in your integer distance set. Every other point in the set is an integer distance from A. The points live in a plane, but you can bump the plane into three-dimensional space by tacking a third coordinate onto each point, whose value is the distance from A. For instance, suppose A is the point (1, 3). Then the point (4, 7), which is 5 units away from A, turns into the point (4, 7, 5) in three-dimensional space. This process converts the plane into a cone in three-dimensional space whose tip sits at A, now labeled (1, 3, 0). The integer distance points become points in three-dimensional space that lie on the cone and also on a certain lattice.

Similarly, if you choose two reference points, A and B, you can convert points in the plane to points in four-dimensional space — just give each point two new coordinates whose values are its distances to A and B. This process converts the plane into a curvy surface in four-dimensional space. You can keep adding more reference points in this way. With each new reference point, the dimension increases by one and the plane gets mapped to an even wigglier surface (or, as mathematicians say, a surface of higher degree).

With this framework in place, the researchers used the determinant method from number theory. Determinants are numbers, usually associated with matrices, that capture a host of geometric properties of a collection of points — for instance, a particular determinant might measure the area of the triangle formed by three of the points. The determinant method offers a way to use such determinants to estimate the number of points that lie simultaneously on a wiggly surface and on a lattice — just the kind of situation Greenfeld, Iliopoulou and Peluse were dealing with.

The researchers used a line of work based on the determinant method to show that when they bump their integer distance set up to a suitably high dimension, the points must all lie on a small number of special curves. These curves, when their shadows in the plane are not a line or a circle, can’t contain many lattice points, which are the only candidates for points in the integer distance set. That means the number of points in the set that can lie off the main line or circle is bounded — the researchers showed that it must be smaller than a very slowly growing function of the set’s diameter.

Their bound doesn’t reach the standard of the “four points off the line or three points off the circle” conjecture that many mathematicians believe is true for large integer distance sets. Even so, the result shows that “the essence of the conjecture is true,” said Jacob Fox of Stanford University. A full proof of the conjecture will likely require another infusion of new ideas, mathematicians said.

The team’s high-dimensional encoding scheme is “extremely robust,” Iosevich said. “There aren’t only applications in principle — there are applications that I’m already thinking about.”

One application, Greenfeld, Iliopoulou and Peluse hope, will be to their original harmonic analysis problem, to which the three are now returning. Their result on integer distance sets “could be a steppingstone toward that,” Greenfeld said.

The synthesis of combinatorics with algebraic geometry that the researchers initiated will not stop with integer distance sets or allied problems in harmonic analysis, Iosevich predicted. “I believe that what we are seeing is a conceptual breakthrough,” he said. “This sends a message to people in both fields that this is a very productive interaction.”

It also sends a message about the value of sometimes making a problem more complicated, Tao said. Mathematicians usually strive for the reverse, he noted. “But this is an example where complexifying the problem is actually the right move.”

The advance has changed the way he thinks about high-degree curves, he said. “Sometimes they can be your friends and not your enemies.”