What does the distance between any two points look like if it is a set of integers? This deceptively simple question has made little progress since Norman Anning and Paul Erdős achieved the results 80 years ago. Now, three mathematicians have made the problem more complex by linking combinatorics, number theory, and algebraic geometry, only to shed light on its proper structure.
撰文 | Erica Klarreich
翻译 | zzllrr小乐
If the points in a large (but not infinite) set of points are integers at a distance from each other, how do these points be arranged? A recent result proves that circles are one of the only possible options. 丨Source: Fran Pulido/Quanta Magazine
The change in plans took place on a road trip. On a sunny day last April, mathematicians Rachel Greenfeld and Sarah Peluse traveled from their home unit, the Institute for Advanced Study in Princeton, New Jersey, to Rochester, New York, where they were both scheduled to give talks the following day.
They have been grappling for nearly two years with an important conjecture in the field of harmonic analysis, which studies how complex signals can be broken down into quantitative frequencies. Together with a third collaborator, Marina Iliopoulou, they investigated the problem of component frequencies being represented as points on a plane, and the distances between these points are related to integers. The three researchers tried to prove that the number of these points could not be too many, but so far, all of their methods have failed.
They seemed to be spinning in circles. Peluse had an idea: what if we gave up on harmonic analysis for a moment and turned our attention to a set of points where the distance between any two points is exactly an integer? What might be the structure of such a set of points? Mathematicians have been trying to understand integer distance sets since ancient times. For example, a Pythagorean array (i.e., a Pythagorean triple, e.g. 3, 4, and 5) represents a right triangle whose distance between the three vertices is an integer.
"Sitting in the car, I guess because Rachel was with me, I came up with the idea," said Peluse, who is now a professor at the University of Michigan. The idea of solving integer distance sets got Greenfeld excited.
Before they realized it, they had gone in the wrong direction twice.
"We didn't actually notice where we were driving or leaving the highway," Peluse said. "We drove about an hour in the opposite direction to Rochester before we noticed because we were so excited. ”
In 1945, Norman Anning and Paul Erdős proved [1] that an infinite set of points on a plane satisfying the distance of any two points that are integers must be on a straight line (collinear). For a finite set of points, the possibilities are more diverse. Mathematicians have constructed large sets of points that sit on a line or circle, sometimes with three or four exceptions that deviate from the main line. (Points themselves don't have to have integer coordinates—they just care that the distance between them is an integer.) ）
Rachel Greenfeld普林斯顿高等研究院数学家丨图源：Andrea Kane
No one has come up with a large set of points with any other structure, but no one has also proven that other structures are impossible. In the nearly 80 years since the publication of Anning and Erdős' results, there has been little progress on the subject – until now.
Greenfeld, Iliopoulou, and Peluse have demonstrated[2] that all the points of a large integer distance set of points—with the possible exception of a few outliers—must be on a straight line or circle. "If you want a large set where all the points are paired and the distance is an integer, then circles and straight lines are the only two possibilities," says József Solymosi of the University of British Columbia. He called their results "brilliant solutions."
This new approach uses ideas and techniques from three different fields of mathematics: combinatorics, number theory, and algebraic geometry. The combination of these different fields "could be a real psychological breakthrough," said Tao Zhexuan, a mathematician at the University of California, Los Angeles.
Alex Iosevich of the University of Rochester agrees. "They have a very strong foundation for a very wide range of issues," he said. "In my opinion, there is no doubt that this will find deeper applications. ”
Limitations of simplicity
On a plane, it's easy to choose an infinite set where all the points are integers at a distance – just take one of your favorite lines, imagine a number axis superimposed on it, and mark all or part of the points corresponding to the integer. But as Anning and Erdős recognized in 1945, this was the only way to construct an infinite set of integer distance points on a plane. Once you have three points that are not on the same line, your configuration becomes limited, making it impossible to add additional points indefinitely.
The reason can be boiled down to simple geometry. Suppose the distance is two points A and B of an integer, and if you want to add a third point C, the distance to A and B is an integer, but not on a straight line that passes A and B, then most points on the plane do not apply. The possible points are on special curves, called hyperbolas, that pass through A and B. If the distance between A and B is 4 units, then there are exactly four such hyperbolas. (Hyperbolas usually have two distinct parts, such as the two red curves below forming a single hyperbola.) ）
图源：Merrill Sherman/Quanta Magazine
Once you've selected C (as shown in the image above, the distance from C to A is 3 units, and the distance to B is 5 units), you have few more options to add more points. Any point you can add must be on some hyperbola between A and B, or on the line through them. But it must also be on a hyperbola between A and C, and on a hyperbola (or a corresponding line) between B and C. In other words, a new point can only be placed at the intersection of three hyperbolas or lines (although not every intersection is applicable). In the beginning, there are only a finite number of these hyperbolas and lines, and two hyperbolas (or lines) intersect at a maximum of four points. So you end up only being able to choose from a finite number of intersections – you can't build an infinite set.
When you want to understand what a finite set of integer distances actually looks like, the hyperbolic approach can quickly become difficult to navigate. As the number of points increases, you have to deal with more and more hyperbolas. For example, when you have 10 points in your set, adding the 11th point will create 10 new hyperbola families - all of these new hyperbolas are between the new points you add and every point already in the set. "You can't add a lot of points because you're going to get lost in all these hyperbolas and intersections," Greenfeld said. ”
As a result, mathematicians are always looking for more manageable ways to construct sets of integer distance points that are not located on a straight line. But they found only one way: to put the dots on a circle. If you want a set of integer distance points with trillions of points, there is a way to find trillions of points on a circle with a radius of 1, and the distances between these points are all fractions. Then you can expand this circle until all fractional distances become integers. The more points you want in the set, the larger the circle needs to be.
Over the years, mathematicians have found only slightly exotic examples. They can construct large sets of integer distance points, all but four of which are in a straight line, or all but three of which are on a circle. Many mathematicians suspect that these are the only large sets of integer distance points, where not all points are on a straight line or a circle. If they can prove the so-called Bombieri-Lang conjecture, they will be convinced of it. But mathematicians disagree on whether this conjecture is possible.
Since the work of Anning and Erdős in 1945, mathematicians have made little progress in understanding integer distance point sets. Over time, integer distance problems seem to be incorporated into a range of other problems in combinatorics, number theory, and geometry, which are simple to understand but seem impossible to solve. "It's a way to measure how pitiful our math is," Tao said. ”
Sarah Peluse,密歇根大学数学家丨图源:Dan Komoda
In a sense, the integer distance problem was a victim of its early success. Hyperbolic proofs, with their ingenious simplicity, are typical of the philosophy that Erdős, a very influential mathematician, often spoke of "The Book" – a hypothetical book that contains the most elegant proofs in mathematics. The culture of simplicity promoted by Erdős has yielded "tremendous results" in combinatorial geometry, says Iosevich. But it can also lead to blind spots – in this case, it's about the value of introducing algebraic geometry methods.
"I don't think you're going to find a result of algebraic geometry in the last 50 years that doesn't involve a lot of technical and complex stuff," Iosevich said. "Sometimes, though, you need to do it. ”
In retrospect, the integer distance problem has been waiting for mathematicians who are willing to consider curves that are more irregular than hyperbolas, and to tame them with esoteric tools from algebraic geometry and number theory. "This requires people with sufficient knowledge and interest," Iosevich said.
Iostevich says that most mathematicians are content to use only a few tools in a certain area of mathematics throughout their careers. But Greenfeld, Iliopoulou and Peluse were intrepid explorers. "They see mathematics as a coherent whole. ”
The problem is complicated
In the summer of 2021, Greenfeld decided it was time to try to solve a harmonized analysis problem that she had been thinking about since graduate school. Classical harmonic analysis, which forms the basis of real-world signal processing, is the core of decomposing signals into sine waves of different frequencies and phases. This process works because an infinite list of sine waves can be derived, and when combined, all the characteristics of any signal can be captured without any redundancy.
However, researchers often want to study something more complex than a one-dimensional signal. For example, they might want to decompose a signal on a disc on a plane. However, the disk can only carry a limited number of compatible sine waves – not enough to capture the behavior of all possible signals on the disk. The question then becomes: how big can this finite set be?
In such a set, the frequency of the sine wave can be represented by points on a plane that seem reluctant to gather in lines and circles: you will never find three points all close to the same line, or all four points close to the same circle. Greenfeld wants to use this exclusion to prove that these frequency sets can only contain a few points.
At a conference at the University of Bonn in 2021, Greenfeld attended a lecture on the "determinant method," a technique from number theory that can be used to estimate how many integer points of certain types can be on a curve. She realized that this tool might be just what she needed. Greenfeld invited Iliopoulou and Peluse, who also attended the meeting. "We started learning this method together," Greenfeld said.
However, despite many efforts, they seem unable to apply the determinant approach to their problems, and by the spring of 2023, they are discouraged. Iosevich invited Greenfeld and Peluse to drive to Rochester for a visit. "So we thought, 'Okay, we're going to Rochester, and talking to Alex is going to get us back on our feet,'" Peluse said. But it turned out that they had regained their spirits by the time they arrived in Rochester by discussing integer distance point sets during an unexpected detour along the Susquehanna River in Pennsylvania.
They arrived and missed the dinner they had planned to have with Iosevich, but they found Iosevich waiting for them in the hotel lobby with a large bag of takeaways. He forgave them for being late, and the next morning, when they told him that they planned to solve the integer distance point set, he was even more tolerant. "He was very excited," Peluse recalls, "and he gave us a huge boost emotionally. ”
Marina Iliopoulou, 崐嶋嶄嶌嶋 The name of the game:Marina Ilioulou
Just like the hyperbolic approach, Greenfeld, Iliopoulou, and Peluse attempt to control the structure of the integer distance point set by identifying the curve family in which the point must be. Once the number of points increases, the hyperbolic approach becomes overly complex, but Greenfeld, Iliopoulou, and Peluse figured out how to account for a larger number of points at the same time by moving the entire configuration to a higher dimensional space.
How does this work, assuming that there is a "reference" point A in the integer distance point set, and every other point in the set has an integer distance from A. These points are on a plane, but you can bounce the plane into 3D space by adding a third coordinate (the value of which is the distance to A) to each point. For example, suppose A is the point (1, 3). Then, a point (4, 7) with a distance of 5 units to A becomes a point (4, 7, 5) in three-dimensional space. This process converts the plane into a cone in three-dimensional space with a vertex of A, which is labeled (1, 3, 0). Integer distance points become points in three-dimensional space, and these points are located on a cone, as well as on a lattice point.
Similarly, if you select two reference points A and B, you can convert the points on the plane into four-dimensional space – just give each point two new coordinates, the value of which is its distance from A and B. This process converts a plane into a surface in a four-dimensional space. You can continue to add more reference points in this way, and with each new reference point, the dimension increases by 1 and the plane is mapped to a more sinuous surface (or what mathematicians call a higher surface).
With this framework, the researchers used a determinant method from number theory. Determinants are numbers associated with matrices that characterize many of the geometric properties of a set of points – for example, a particular determinant might measure the area of a triangle formed by three points. The determinant method provides a way to estimate the number of points that lie on both a sinuous surface and a lattice point – which is exactly what Greenfeld, Iliopoulou, and Peluse are dealing with.
Using a series of work based on the determinant method, they proved that when the set of integer distance points is elevated to an appropriately high dimension, the points must all be on a handful of special curves. These curves, when their shadows are not straight lines or circles on the plane, cannot contain many lattice points that are the only candidates for points in the integer distance point set. This means that there is a limit to the number of points in the set that may be outside the main line or circle – the researchers proved that this number must be less than a very slowly growing function on the diameter of the set.
Their boundaries do not meet the standard of the "four-point offline or three-point off-circle" conjecture that many mathematicians consider to be correct for large integer distance point sets. Even so, this result suggests that "the nature of the conjecture is correct," said Jacob Fox of Stanford University. Mathematicians say a complete proof of this conjecture may require an injection of new ideas.
Iosevich said the team's high-dimensional coding scheme is "extremely robust." "It's not just in principle, I'm already thinking about practical applications," he says. ”
Greenfeld, Iliopoulou, and Peluse hope that their original harmonic analysis problem will be an application. They are now back to the question. Their results on integer distance point sets "could be a stepping stone to that problem," Greenfeld said.
Iosevich predicts that researchers are beginning to combine combinatorics with algebraic geometry, and that this combination will not stop at integer distance point sets or related problems in harmonic analysis. "I believe what we're seeing is a conceptual breakthrough," he said. "This sends a message to mathematicians in both fields that this interaction is very productive. ”
Tao said it also conveys a message that sometimes makes things more complex is valuable. He points out that mathematicians often pursue opposite goals. "But this is an example where complicating the problem is actually the right thing to do. ”
He said the development has changed his view of higher-order curves. "Sometimes, they can be your friends, not enemies. ”
bibliography
[1] https://www.ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/
[2] https://arxiv.org/abs/2401.10821
本文经授权转自“zzllrr小乐”公众号，原标题《小乐数学科普：融合领域，数学家们在老问题上走得更远》；《返朴》对译文进行了校订。 本文译自Merging Fields, Mathematicians Go the Distance on Old Problem，原文链接：https://www.quantamagazine.org/merging-fields-mathematicians-go-the-distance-on-old-problem-20240401/。
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