内容来源：https://www.quantamagazine.org/networks-hold-the-key-to-a-decades-old-problem-about-waves-20260128/

内容总结：

数学界数十年难题获突破：图论工具破解傅里叶分析经典猜想

一项困扰数学家近六十年的傅里叶分析核心难题，近期意外地通过图论方法取得了关键进展。这项突破由四位年轻数学家——金志涵、米洛耶维奇、托蒙和张盛桐——共同完成，他们首次将傅里叶级数的最小值问题转化为图论中的“最大割”问题，并借助图结构分析取得了里程碑式的成果。

该难题源于1965年数学家萨瓦达曼·乔拉提出的“乔拉余弦问题”：对于任意N个正整数构成的集合，其对应余弦波之和（一种最简单的傅里叶级数）的最小值能否低于负的根号N？尽管问题表述简洁，却深刻揭示了数学家对傅里叶级数深层结构的认知局限。过去数十年来，相关研究进展缓慢，最佳结果始终停留在2004年数学家伊姆雷·鲁萨证明的常数下界，与乔拉的预测相去甚远。

转机出现在2023年夏季。当时，两组图论研究者正分别攻关图论中的经典“最大割”问题。他们通过分析图的负特征值与最大割之间的关联，得出了一项关键结论：若一个图没有大团（即紧密连接的节点簇），则其最小特征值必然很小。这一发现被数论专家什克雷多夫敏锐捕捉到——他意识到，乔拉问题中余弦和的最小值恰好等价于一类特殊“凯莱图”的最小特征值。而凯莱图的结构天然排斥大团，因此可直接应用上述图论结论。

托蒙在后续推导中找到了关键矛盾点：假设凯莱图的最小特征值不低（即余弦和不小），则根据图论结论，图中必存在大团；但凯莱图的代数构造严格限制了边的数量与分布，大团的存在将导致边数超出可能范围，从而反证最小特征值必须足够小。基于此，团队最终证明：对于任意N个整数集，其余弦和最小值至少低于负的N的1/10次方。这一结果首次以与乔拉猜想相同的幂律形式给出了下界，突破了长期停滞的常数边界。

几乎同时，剑桥数学家贝德特通过传统傅里叶分析方法将下界进一步提升至负的N的1/7次方。两项独立工作共同标志着该问题研究范式的转变：图论工具的引入为傅里叶分析打开了新视野。

牛津数学家桑德斯评价道：“这有点像登月或突破四分钟英里——我们无法预知它将开启什么。”研究者们相信，这一跨领域连接并非偶然，它暗示了傅里叶分析与图论之间可能存在更深刻的统一性。正如哥伦比亚大学教授索尼所言：“知道这些事物处于同一世界是非常有力的信息。”未来，这一思路有望在更多数学领域激发新的突破。

中文翻译：

网络结构成为破解数十年波动难题的关键

引言

两个世纪前，约瑟夫·傅里叶为数学家们提供了一种神奇的方法。他猜想，几乎任何函数都可以写成一系列简单波动的叠加，这种技巧如今被称为傅里叶变换。如今，傅里叶变换被用于理解从遥远恒星的化学成分到地壳深处发生的一切。

"傅里叶级数在数学中无处不在，"哥伦比亚大学的梅塔布·索内表示，"傅里叶级数很重要，这几乎是数学家们的信仰。"

然而，关于傅里叶变换的一些基本问题，却一直顽固而神秘地悬而未决。

1965年，数学家萨瓦达曼·乔拉提出了这样一个问题。他想知道一种极其简单的傅里叶变换——余弦波之和——的最小值能有多小。他的问题听起来很直接，但不知为何，却并非如此。

"这个问题有点像诱饵，"索内说；它的设计初衷就是为了揭示数学家所知是多么有限。"因为我们无法证明这一点，显然我们完全不了解这些[和]的结构。"

数十年来，数学家们一直在努力攻克乔拉的余弦问题。它成为了傅里叶分析技术的一个基准，用于探索这些技术能在多大程度上探测数字序列的深层结构。结果令人沮丧。"进展完全停滞不前，"牛津大学的汤姆·桑德斯说。

今年九月，情况突然发生了变化。四位数学家——金志涵、阿莱克萨·米洛耶维奇、伊什特万·托蒙和张盛桐——在该问题上取得了20年来的首次重大进展。他们的策略与传统傅里叶分析几乎毫无关系。

事实上，在去年夏天之前，这四人甚至从未听说过乔拉的余弦问题。

低谷探秘

20世纪50年代初，乔拉和他的数论同行内史密斯·安肯尼想利用傅里叶变换来更好地理解数字集合中的模式。考虑由数字2、3和8组成的集合。首先，用集合中的每个数字定义一个余弦波——例如，2对应cos(2x)。然后将所有波相加，得到cos(2x) + cos(3x) + cos(8x)。这只是将原始集合写成傅里叶级数的另一种方式。这个级数结构非常规整：所有波都是余弦波，并且由于没有任何余弦波前面有系数，所有波的振幅都相同。"这是你能得到的最简单的傅里叶级数类型，"剑桥大学的本杰明·贝德特说。"总的来说，我们对傅里叶级数了解颇多。"

由cos(2x) + cos(3x) + cos(8x)定义的新波具有峰谷，揭示了原始数字集合的有趣性质。因此，安肯尼和乔拉试图测试他们对这种级数的理解程度。他们想知道：对于任意由N个整数组成的集合，这个和所能达到的最小值是多少？

求这个和的最大值很容易。当x为零时，任何余弦波都达到其最大值1。因此，我们三个余弦波的和是1 + 1 + 1，即3。同样，一千万个余弦波的和最大值就是一千万。对于任意N个整数的集合，最大值就是N。

然而，理解余弦和的最小值却出奇地困难。虽然不同的波至少在一点（当x为零时）同时达到最大值，但对于最小值却并非如此。也许不同波的低谷仍然会足够对齐，从而产生一个非常低的和。或者，这些波可能会相互干扰，使得和不可能变得太低。

从左至右：金志涵提供；上沃尔法赫数学研究所档案馆；利维亚·托蒙-霍尔瓦特

1952年，安肯尼和乔拉猜想，正如最大值随着原始集合中整数数量的增加而越来越高一样，最小值也应该越来越低。几年后，这一点得到了证明——这促使乔拉在1965年将问题进一步精确化。他想确切知道最小值随着N的增长下降得有多快。

他知道存在一些N个整数的集合，其余弦和的最小值大约为−$latex \sqrt{\textit{N}}$。他能想到的其他所有集合的下降幅度甚至更低，这使他猜想：对于任意由N个正整数组成的集合，相应余弦和的最小值必须低于−$latex \sqrt{\textit{N}}$。

在随后的几十年里，一些数学家逐步推进了这个问题。但到了21世纪初，他们能够证明的结果与乔拉的预测之间仍然存在巨大鸿沟。根据2004年匈牙利阿尔弗雷德·雷尼数学研究所的伊姆雷·鲁萨证明的最新界限，一个由10^20个余弦组成的和——也就是1后面跟着20个零，大约是一立方英寸空气中的分子数——其最小值必须小于大约−7。相比之下，乔拉预测最小值必须低于−10^10。

然而，在过去的20年里，鲁萨的结果一直代表着乔拉余弦问题进展的顶峰。

直到一个完全无关的研究项目最终打破了这一壁垒。

架起桥梁

该项目涉及一种由节点和边组成的网络，称为图。

去年夏天，两组图论研究者——在欧洲的金志涵、米洛耶维奇和托蒙，以及在斯坦福大学的张盛桐——正热情高涨地推进图论中最核心的问题之一。"最大割"问题涉及将图最优地切割成两部分，使得连接两部分的边尽可能多。这是一个关于图结构的基本问题，具有实际应用：例如，一个图的最大割可能代表高效的电路设计，或粒子系统的最低能量状态。

马克·贝兰/塞缪尔·维拉斯科/《量子》杂志

但目前还没有一种放之四海而皆准的方法来寻找图的最大割。（这是一个所谓的NP难问题。）因此，数学家们转而试图估计特定类别图的最大割。

2003年，苏黎世联邦理工学院的数学家本杰明·苏达科夫（他后来指导了金志涵、米洛耶维奇和托蒙）提出了关于特定类型图的最大割的一个猜想。这种图没有团——即所有节点都相互连接的节点簇。

去年七月，二十多年后，张盛桐证明了这类图最大割的一个新界限。几天后，金志涵、米洛耶维奇和托蒙改进了他的结果。

为此，研究人员研究了称为特征值的重要量。特征值提供了关于图结构的信息。例如，最大特征值计算图中的边数；第二大特征值衡量图的连通性。金志涵、米洛耶维奇、托蒙和张盛桐专注于负特征值，基于最近一系列将负特征值与图的最大割联系起来的研究。他们对这些特征值的分析最终使他们能够证明新的结果。

数学家们决定将他们各自的结果合并成一篇联合论文。但在他们完成之前，他们收到了一封关于乔拉余弦问题的意外邮件。

凯莱图

邮件来自印第安纳州普渡大学的数学家伊利亚·什克雷多夫。作为数论学家，什克雷多夫指出，乔拉的余弦问题可以用图来重新表述。不是团队正在研究的一般类型的图，而是由数学家亚瑟·凯莱在1878年发明的一种特殊类型的图。

要构建一个凯莱图，想象你再次处理集合{2, 3, 8}。从一堆节点开始——节点数量并不重要，只要节点数是质数且大于集合中最大的整数即可。接下来，将节点排列成一个圆圈，并为每个节点标记一个整数。然后，如果两个节点之间的差值在原始集合中，就在它们之间连一条边。因此，标记为1和3的节点将由一条边连接，因为它们的差是2，而2在集合{2, 3, 8}中。

到了20世纪70年代，数学家们已经发现，凯莱图的结构中嵌入了来自乔拉问题的傅里叶级数的信息。事实证明，凯莱图的特征值正好对应于余弦和可以取的不同值。因此，最小特征值告诉你余弦和能有多低。

"这是众所周知的事情，"米洛耶维奇说。"这种联系非常经典。"

它使数学家能够重新表述问题。如果他们能证明凯莱图的最小特征值变得非常小，那就意味着余弦和也必须变得非常小——这正是乔拉余弦问题的核心所在。

但没人能想出如何利用这种联系。

"只有当你有了锤子，你才会尝试用锤子敲钉子，"苏达科夫说。数学家们没有足够精确地分析最小特征值的方法，以找出他们想知道的关于余弦和最小值的信息。

万琪·朱

但在他们关于图的最大割的研究中，金志涵、米洛耶维奇、托蒙和张盛桐无意中打造了一把锤子。在研究图的特征值与其结构的关系时，他们发现任何没有低特征值的图必然由团主导。什克雷多夫在阅读他们的证明时意识到，这意味着该团队实际上再次重新表述了乔拉的余弦问题：不再需要直接分析特征值。相反，他们只需要证明凯莱图没有任何大团。这将意味着这些图各自都有一个非常低的特征值，最终使他们能够利用乔拉猜想和图论之间的联系。

从那时起，"我认为主要的障碍是相信我们能做到，"托蒙说。

团结构豁然开朗

当什克雷多夫发送邮件时，数学家们都在度假。但正在家乡布达佩斯访问的托蒙，抽时间研究了凯莱图。

思考了一会儿后，"突然就豁然开朗了，"他说。

要理解托蒙的想法如何运作，让我们回到集合{2, 3, 8}的凯莱图。记住，证明乔拉猜想意味着证明图的最小特征值变得非常低。所以首先假设相反的情况：没有任何特征值是低的。你将希望证明这个假设最终会导致矛盾。

根据团队在最大割方面的工作，如果凯莱图没有小的特征值，那么它必须有一个大团——比如，五个节点全部相互连接。这反过来意味着，如果你取这些节点中的任意两个，它们的整数标签之间的差是2、3或8。

罗曼娜·米雷斯

但现在给每个节点加1，得到一组新的五个节点。它们之间的差将与第一组相同，这意味着它们也将形成一个团。继续下去，你会生成越来越多的团。但有一个问题：团有很多边，而凯莱图根据其定义，边相对较少，并且遵循非常特殊的结构。最终，你会得到如此多的团，以至于生成的边数超过了凯莱图所能容纳的。这意味着之前存在一个大团的假设一定是错误的。这反过来又意味着最小特征值必须很低。

一旦托蒙想通了这一点，证明的其余部分就相对容易地组合起来了。九月，他、金志涵、米洛耶维奇和张盛桐将他们的联合论文在线发布。论文主要关注如何分析图的最小特征值——这项工作一方面使他们能够加强几个月前在无团图的最大割方面发现的界限。

但他们最重要的成果是关于乔拉的余弦问题。他们证明了对于任意由N个整数组成的集合，相应的余弦和会达到一个低于−N^(1/10)的值。对于任何现实的N值，−N^(1/10)与鲁萨几十年前的界限相差不大。但对于巨大的N值，比如10^20，差异开始变得明显：金志涵、米洛耶维奇、托蒙和张盛桐证明，10^20个余弦的和会低于−100，而鲁萨的界限是−7。

"对我来说，这非常令人惊讶，"苏达科夫说。这个团队从一个关于图的结果开始，却意外地获得了一个看似无关问题的新见解。

就在研究人员发布论文两天后，剑桥大学的数学家贝德特使用傅里叶分析中更传统的方法，在该问题上取得了自己的进展。他的结果以微弱的优势超过了团队的界限：它表明对于任意由N个整数组成的集合，余弦和会达到一个小于−N^(1/7)的值。对于10^20，这将金志涵、米洛耶维奇、托蒙和张盛桐确定的最小值从−100降低到大约−720。

但数学家们认为最值得注意的是，这两个结果都标志着首次有一个被证明的估计具有与乔拉猜想界限相同的形式。也就是说，新的界限，像乔拉的一样，可以写成N的幂。（乔拉的界限−$latex \sqrt{\textit{N}}$等价于−N^(1/2)。）鲁萨之前的估计无法写成这种形式。

围绕傅里叶变换的迷雾依然浓密。但这些新技术能稍微更好地看透它。

尽管两个证明都尚未完全弥合差距以证明乔拉猜想，但数学家们感到兴奋。就目前而言，"我认为这有点像登月或四分钟跑一英里，"桑德斯说。"目前还不清楚这将开启什么。"

图在这个故事中扮演的角色尤其引人入胜。这不是图论和傅里叶分析第一次相遇。但到目前为止，这两个领域之间的联系都是孤立的。现在，金志涵希望乔拉余弦问题与最大割之间的具体联系暗示着更广泛的东西。"乔拉问题中预测的任何现象都更具普遍性，"他说。"它在图中也适用。"

"我们现在有更多问题处于相同的影响范围内，"索内说。"知道事物存在于同一个世界是非常有用的信息。这非常强大。"

英文来源：

Networks Hold the Key to a Decades-Old Problem About Waves
Introduction
Two centuries ago, Joseph Fourier gave mathematicians a magical technique. He conjectured that it’s possible to write almost any function as a sum of simple waves, a trick now called the Fourier transform. These days, the Fourier transform is used to understand everything from the chemical makeup of distant stars to what’s happening far beneath the Earth’s crust.
“Fourier series are everywhere in mathematics,” said Mehtaab Sawhney of Columbia University. “It’s part of the faith of mathematicians that Fourier series are important.”
Yet certain fundamental questions about the Fourier transform have remained stubbornly, and mysteriously, unanswerable.
In 1965, the mathematician Sarvadaman Chowla posed one such question. He wanted to know how small an extremely simple type of Fourier transform — a sum of cosine waves — could get. His problem sounded straightforward. But somehow, it wasn’t.
“The question is a bit of bait,” Sawhney said; it was designed to illuminate just how little mathematicians know. “Because we can’t show this, we clearly don’t understand the structure of these [sums] at all.”
For decades, mathematicians struggled with Chowla’s cosine problem. It became a benchmark for Fourier analysis techniques, used to explore how well they could detect deeper structure in sequences of numbers. The results were discouraging. “Progress was completely anemic,” said Tom Sanders of the University of Oxford.
In September, that suddenly changed. Four mathematicians — Zhihan Jin, Aleksa Milojević, István Tomon, and Shengtong Zhang — posted the first significant advance on the problem in 20 years. Their strategy had almost nothing to do with traditional Fourier analysis.
In fact, before last summer, the foursome had never even heard of Chowla’s cosine problem.
Feeling Low
In the early 1950s, Chowla and his fellow number theorist Nesmith Ankeny wanted to use the Fourier transform to better understand patterns in sets of numbers. Consider the set consisting of the numbers 2, 3, and 8. First, use each number in the set to define a cosine wave — 2 gives you cos(2x), for instance. Then add up all your waves to get cos(2x) + cos(3x) + cos(8x). This is just another way of writing your original set as a Fourier series. The series is highly structured: All the waves are cosines, and because there are no numbers in front of any of the cosines, all the waves are the same size. “It’s the simplest possible type of Fourier series you can have,” said Benjamin Bedert of the University of Cambridge. “And in general, we know quite a lot about Fourier series.”
The new wave defined by cos(2x) + cos(3x) + cos(8x) has peaks and valleys that reveal interesting properties of the original set of numbers. So Ankeny and Chowla sought to test how much they really understood about such a series. They wondered: For any set of N integers, what is the lowest value that the sum will ever take?
It’s easy to figure out the sum’s maximum. When x is zero, any cosine wave hits its maximum at 1. So our sum of three cosine waves gives us 1 + 1 + 1, or 3. Similarly, a sum of 10 million cosine waves has a maximum of 10 million. For any set of N integers, the maximum is simply N.
Yet understanding the cosine sum’s minimum is surprisingly difficult. While the different waves all hit their maximum simultaneously at least once (when x is zero), this is not true for the minimum. Perhaps the lowest points of the different waves will still align enough to produce a very low sum. Or perhaps the waves will interfere with each other so that it becomes impossible for the sum to get too low.
From left: Courtesy of Zhihan Jin; Archives of the Mathematisches Forschungsinstitut Oberwolfach; Livia Tomon-Horvath
In 1952, Ankeny and Chowla conjectured that just as the maximum gets higher and higher as the number of integers in the original set gets bigger, the minimum should get lower and lower. This was proved several years later — prompting Chowla to sharpen the question in 1965. He wanted to know exactly how fast the minimum drops as N grows.
He knew of sets of N integers whose cosine sum had a minimum value around −$latex \sqrt{\textit{N}}$. Every other set he could think of dipped even lower, leading him to conjecture that for any set of N positive integers, the minimum of the corresponding cosine sum must be below −$latex \sqrt{\textit{N}}$.
Over the ensuing decades, a few mathematicians chipped away at the problem. But by the mid-2000s, there was still a massive gulf between what they were able to prove and what Chowla had predicted. According to the latest bound, proved in 2004 by Imre Ruzsa of the Alfréd Rényi Institute of Mathematics in Hungary, a sum of 1020 cosines — that’s a 1 with 20 zeros after it, about the number of molecules in a cubic inch of air — must have a minimum value smaller than about −7. By comparison, Chowla had predicted that the minimum would have to dip below −1010.
And yet, for the past 20 years, Ruzsa’s result has represented the pinnacle of progress on Chowla’s cosine problem.
Then an entirely unrelated research program finally broke the barrier.
Bridging the Divide
The program dealt with networks of nodes and edges called graphs.
Last summer, two sets of graph theorists — Jin, Milojević, and Tomon in Europe, and Zhang at Stanford University — were enthusiastically making progress on one of graph theory’s most central questions. The “MaxCut” problem is about the optimal way to cut a graph into two parts so that there are as many edges as possible connecting the parts. It’s a basic question about the structure of a graph, with real-world applications: A graph’s MaxCut might represent an efficient circuit design, for instance, or the lowest-energy state of a system of particles.
Mark Belan/Samuel Velasco/Quanta Magazine
But there’s no one-size-fits-all approach to finding a graph’s MaxCut, at least at the moment. (It’s what’s known as an NP-hard problem.) And so mathematicians instead attempt to estimate the MaxCut for specific classes of graphs.
In 2003, Benjamin Sudakov, a mathematician at the Swiss Federal Institute of Technology Zurich who would later mentor Jin, Milojević, and Tomon, posed a conjecture about the MaxCut of a particular kind of graph. This graph had no cliques — clusters of nodes that are all connected to one another.
Last July, more than two decades later, Zhang proved a new bound on the MaxCut for such graphs. A few days later, Jin, Milojević, and Tomon improved on his result.
To do this, the researchers investigated important quantities called eigenvalues. Eigenvalues provide information about a graph’s structure. For example, the largest eigenvalue counts the number of edges in the graph; the second-largest measures the graph’s connectivity. Jin, Milojević, Tomon, and Zhang focused on the negative eigenvalues, building on a recent line of research that had linked them to a graph’s MaxCut. Their analysis of these eigenvalues ultimately allowed them to prove their new results.
The mathematicians decided to combine their separate results into a joint paper. But before they could finish, they received an unexpected email about Chowla’s cosine problem.
Cayley’s Graph
The email was from Ilya Shkredov, a mathematician at Purdue University in Indiana. Shkredov, a number theorist, pointed out that Chowla’s cosine problem could be reformulated in terms of graphs. Not the general kinds of graphs that the team was studying, but a special type of graph invented in 1878 by the mathematician Arthur Cayley.
To build a Cayley graph, imagine you’re once again working with the set {2, 3, 8}. Start with a bunch of nodes — it doesn’t really matter how many, so long as the number of nodes is prime and larger than the biggest integer in the set. Next, arrange the nodes in a circle and label each one with an integer. Then place an edge between two nodes if the difference between them is in the original set. So the nodes labeled 1 and 3 will be connected by an edge, because they differ by 2, and 2 is in the set {2, 3, 8}.
By the 1970s, mathematicians had figured out that embedded within the structure of Cayley graphs is information about the Fourier series from Chowla’s problem. A Cayley graph’s eigenvalues, it turns out, correspond exactly to different values that the cosine sum can have. The smallest eigenvalue therefore tells you how low the cosine sum can get.
“It’s a well-known thing,” Milojević said. “The connection is very classical.”
It allowed mathematicians to reframe the problem. If they could show that the smallest eigenvalue of a Cayley graph gets very small, it would mean that the cosine sum has to get very small as well — precisely what Chowla’s cosine problem is all about.
But no one could figure out how to exploit that connection.
“You try to hit a nail with a hammer only once you have a hammer,” Sudakov said. Mathematicians didn’t have a way of analyzing the lowest eigenvalue accurately enough to find out what they wanted to know about the minimum of the cosine sum.
Wanqi Zhu
But in their work on the MaxCut of graphs, Jin, Milojević, Tomon, and Zhang had unwittingly produced a hammer. While studying how the eigenvalues of a graph relate to its structure, they’d discovered that any graph that doesn’t have a low eigenvalue must be dominated by cliques. Shkredov, reading their proof, realized that this meant that the team had actually reframed Chowla’s cosine problem once more: There was no longer any need to analyze the eigenvalue directly. Instead, they just had to prove that the Cayley graphs didn’t have any large cliques. That would imply that the graphs each had a very low eigenvalue, finally enabling them to exploit the link between Chowla’s conjecture and graph theory.
From then on, “I think the main obstacle was believing we can do it,” Tomon said.
The Cliques Click
When Shkredov sent his email, the mathematicians were all on vacation. But Tomon, who was visiting his home city of Budapest, found time to toy with the Cayley graph.
After a bit of thinking, “it just clicked,” he said.
To see how Tomon’s idea works, let’s go back to our Cayley graph for the set {2, 3, 8}. Remember that proving Chowla’s conjecture means showing that the graph’s smallest eigenvalue gets very low. So first assume the opposite: that none of the eigenvalues are low. You’ll want to show that this assumption will eventually lead to a contradiction.
Based on the team’s work on MaxCut, if the Cayley graph has no small eigenvalues, then it must have a large clique — say, five nodes that are all connected to each other. This, in turn, means that if you take any two of those nodes, the difference between their integer labels is 2, 3, or 8.
Romana Meereis
But now add 1 to each node to get a new set of five nodes. They’ll differ by the same amounts as the first set, meaning that they, too, will form a clique. Keep going, and you’ll generate more and more cliques. But there’s a problem: Cliques have lots of edges, while a Cayley graph, based on how it’s defined, has relatively few edges, which obey a very particular structure. Eventually, you’ll get so many cliques that you’ll have generated more edges than the Cayley graph can hold. This means that the earlier assumption that there was a large clique must have been false. Which, in turn, means that the smallest eigenvalue had to be low.
Once Tomon figured this out, the rest of the proof came together relatively easily. In September, he, Jin, Milojević, and Zhang posted their joint paper online. It mainly focused on how to analyze the lowest eigenvalues of graphs — work that, for one thing, allowed them to strengthen the bounds they’d found a few months earlier on the MaxCut of graphs without cliques.
But their headline result was about Chowla’s cosine problem. They’d proved that for any set of N integers, the corresponding cosine sum attains a value lower than −N1/10. For any realistic value of N, −N1/10 doesn’t differ too much from Ruzsa’s decades-old bound. But for huge values of N, like 1020, the difference starts to be noticeable: Jin, Milojević, Tomon, and Zhang show that a sum of 1020 cosines slides below −100, in comparison to Ruzsa’s bound of −7.
“For me, it’s very surprising,” Sudakov said. The group started with a result about graphs, and out of nowhere, they gained fresh insight on a seemingly unrelated problem.
Just two days after the researchers posted their paper, Bedert, the Cambridge mathematician, posted his own advance on the problem, using a more traditional approach from Fourier analysis. His result edges out the team’s bound by a hair: It says that for any set of N integers, the cosine sum attains a value less than −N1/7. For 1020, this lowers the minimum that Jin, Milojević, Tomon, and Zhang identified from −100 to around −720.
But what mathematicians find most noteworthy is that both of these results mark the first time that a proven estimate has the same form as Chowla’s conjectured bound. That is, the new bounds, like Chowla’s, can be written as a power of N. (Chowla’s bound of −$latex \sqrt{\textit{N}}$ is equivalent to −N1/2.) Ruzsa’s previous estimate cannot be written in this form.
The fog surrounding the Fourier transform is still dense. But these new techniques are a little better at seeing through it.
Though neither proof has fully bridged the gap to prove Chowla’s conjecture, mathematicians are excited. For now, “it is a little bit, I think, like the moon landing or the 4-minute mile,” Sanders said. “It’s not clear ahead of time what this is going to open up.”
The role that graphs played in the story is particularly intriguing. This isn’t the first time that graph theory and Fourier analysis have met. But so far, the links between the two fields have been one-offs. Now, Jin hopes that the specific connection between Chowla’s cosine problem and MaxCut hints at something broader. “Whatever is predicted in the Chowla problem, that phenomenon is more general,” he said. “It works in graphs.”
“We now have more problems that are in the same spheres of influence,” Sawhney said. “Knowing that things are living in the same world is very useful information. It’s very powerful.”