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Selected from quantamagazine

Compiled by Machine Heart
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Recently, a mathematical problem that has remained unsolved for decades has made progress for the first time.

The researchers who drove this progress are James Leng, a graduate student from UCLA, Ashwin Sah, a mathematics graduate student from MIT, and Mehtaab Sawhney, an assistant professor at Columbia University. James Leng studied under the famous mathematician Terence Tao, and Ashwin Sah studied under the discrete mathematics expert Yufei Zhao.

Paper address: https://arxiv.org/pdf/2402.17995

To understand the breakthrough achieved in this research, we need to start with arithmetic series.

The sum of the first n terms of an arithmetic progression is called an arithmetic series, also called an arithmetic series. In 1936, mathematicians Paul Erdős and Pál Turán conjectured that if a set consists of non-zero fractions of integers (even 0.00000001%), then it must contain an arithmetic series of arbitrary length. The only sets that can avoid arithmetic series are those that contain a "negligible" fraction of integers. For example, the set {2, 4, 8, 16, ...}, where each number is twice the previous number, is scattered along the number line and has no series.

In 1975, mathematician Endre Szemerédi proved the conjecture. His work spawned many research directions that mathematicians are still exploring today.

Mathematicians established Szemerédi’s result in the case of finite sets of numbers: all integers from 1 to some number N. What fraction of the initial pool is available to be used before inevitably containing a forbidden series? How does this fraction change as N changes?

For example, let N be 20. How many of these 20 numbers can be written down while still avoiding series of length 5 or more numbers? It turns out that the answer is between 16% and 80% of the initial pool.

Szemerédi was the first to show that this fraction must shrink to zero as N grows, and mathematicians have since tried to quantify how quickly this happens.

Last year, breakthrough work by two computer scientists almost solved the problem for series with three terms, such as {6, 11, 16}. But the problem becomes more difficult when you try to avoid arithmetic series with four or more terms. That’s because longer series reveal underlying structures that are difficult to uncover using classical mathematical methods.

It’s relatively easy to prove whether a set contains numbers that always satisfy the simple equation x – 2y + z = 0, which is the case with the series {10, 20, 30} (10 – 2*(20) + 30 = 0). In contrast, numbers in a series with four terms must also satisfy the more complex equation x^2 – 3y^2 + 3z^2 – w^2 = 0, and series with five or more terms must satisfy even more complex equations. This means that sets containing such series can exhibit more subtle patterns. It’s also harder for mathematicians to prove whether such patterns exist.

In the late 1990s, mathematician Timothy Gowers proposed a theory that overcomes this obstacle. He was later awarded the Fields Medal, mathematics’ highest honor, in part for this work. In 2001, he applied his method to Szemerédi’s theorem and proved a better bound on the maximum size of a set that avoids arithmetic progressions of any given length.

In 2022, James Leng, then a second-year graduate student at UCLA, began to understand Gowers' theory. He wasn't thinking about Szemerédi's theorem. Instead, he wanted to answer questions related to Gowers's method.

However, after more than a year of hard exploration, he found nothing.

Sah and Sawhney, who had been thinking about related issues, learned about Leng's work and were very interested. Sawhney even said, "I'm surprised that I can think this way."

Sah and Sawhney realized that Leng’s work might help them make further progress on Szemerédi’s theorem. Within a few months, the three young mathematicians figured out how to get a better upper bound on the size of a set without the five-term series. They then extended their work to series of arbitrary lengths, marking the first progress on the problem in the 23 years since Gowers’ proof.

express

, the size of the largest subset that does not have a k-term arithmetic progression. Leng, Sah, and Sawhney showed that for k ≥ 5, there exists c_k > 0 such that

Research Team

The first author of the paper, James Leng, is a mathematics graduate student at the University of California, Los Angeles (UCLA) and received his undergraduate degree from the University of California, Berkeley. He studied under the famous mathematician Terence Tao.

James Leng's research interests include arithmetic combinatorics, dynamical systems, and Fourier analysis, etc. His research has also been supported by NSF graduate student fellowships.

James Leng

Ashwin Sah has loved mathematics since he was a child. He was exposed to advanced mathematics in competitions and performed well. In the summer of 2016, the 16-year-old Sah won the gold medal in the International Mathematical Olympiad (IMO), and the following year he entered MIT to study.

Ashwin Sah

During his time at MIT, two people played an important role in Sah's mathematical development. The first was Professor Yufei Zhao, a master of discrete mathematics and Sah's graduate advisor.

The second is Mehtaab Sawhney, whom they met in class and became friends with. Later, the two did research together, exploring multiple topics in the field of discrete mathematics, such as graph theory, probability theory, and the properties of random matrices. Ashwin Sah and Mehtaab Sawhney met as undergraduates at MIT in late 2017. Since then, the two have written an incredible 57 mathematical proofs together, many of which have had a profound impact in various fields.

Mehtaab Sawhney

Mehtaab Sawhney is currently an assistant professor at Columbia University. His research interests include combinatorics, probability, and theoretical computer science.

参考链接：https://www.quantamagazine.org/grad-students-find-inevitable-patterns-in-big-sets-of-numbers-20240805/