The new formula showed how deeply mathematicians had been mistaken about simple waves.
Two hundred years ago, Joseph Fourier proposed an idea that would eventually become the foundation of modern mathematics and physics. He proposed that almost any function can be represented as a sum of simple waves. Today, this principle, which underlies the Fourier transform , is used in analyzing the spectra of distant stars, processing signals, investigating the structure of matter, and studying processes occurring deep beneath the Earth's crust . Despite the universality of the method, questions remain surrounding it that have defied rigorous mathematical analysis for decades.
One such problem was formulated in 1965 by mathematician Sarvadaman Chowla. He became interested in an extremely simple but fundamental object: the sum of cosines without coefficients, where each function has the same amplitude and differs only in frequency. Formally, this is an expansion of the form cos(a₁x) + cos(a₂x) + ... + cos(aₙx), where a₁, a₂, …, aₙ are integers. From the perspective of Fourier theory, this is one of the most elementary types of series, but it turned out to be unexpectedly difficult to analyze.
The maximum value of such a sum is determined trivially. At x = 0, each cos(ax) function is equal to 1, so the sum of N cosines always reaches the value N. The minimum, by contrast, is much more complex. The minimum points of individual waves do not coincide in time, and the behavior of the sum is determined by complex interference of different frequencies. A fundamental question arises: how deeply can the graph of such a sum descend into negative territory?
Chowle knew of examples of sets of N integers for which the minimum sum approached a value of the order of -√N. He also observed that most other sets yielded even more negative values. This led him to the conjecture that for any set of N positive integers, the corresponding cosine sum necessarily takes a value below -√N, as well as to the more precise question of how quickly this lower bound decreases as N increases.
Despite the simplicity of its formulation, the problem remained virtually unsolved for decades. In 2004, Imre Ruzsa obtained a result that was long considered the best known. His estimate provided an extremely weak lower bound on large scales. For example, for a sum of 10²⁰ cosines, it guaranteed a value below approximately -7, while Chowla's conjecture suggested a level of about -10¹⁰. For nearly 20 years, this estimate remained the pinnacle of progress.
The shift occurred unexpectedly and came from a completely different area of mathematics. Four researchers—Zhihan Jin, Aleksa Milojevic, Istvan Tomon, and Shengtong Zhang—were studying graph theory problems, specifically the MaxCut problem. It involves optimally partitioning a graph into two parts so as to maximize the number of edges between them. This problem has both theoretical significance and applied interpretations in physics, computer science, and engineering, but is NP-hard and lacks a universal algorithmic solution.
In their research, the team analyzed the spectral characteristics of graphs, that is, the eigenvalues of the matrices describing their structure. These quantities reflect fundamental properties such as connection density, connectivity, and the presence of clusters. Particular attention was paid to negative eigenvalues, as they were found to be closely related to MaxCut scores. The study yielded a general result: if a graph lacks sufficiently small negative eigenvalues, its structure is inevitably dominated by cliques—dense subgraphs where every vertex is connected to every other.
A further turnaround occurred after a letter from Ilya Shkredov, who pointed out the connection between Chowla's problem and a special class of objects—Cayley graphs. These graphs are constructed over a set of integers and a prime modulus, and their spectral characteristics are directly related to the values of the corresponding cosine sums. The minimum eigenvalue of such a graph coincides with the smallest value of the cosine sum.
Using previously obtained results on MaxCut, the researchers were able to reformulate the problem. Instead of a direct analysis of the eigenvalues, it was now sufficient to prove that large cliques cannot exist in the corresponding Cayley graphs. The assumption of their existence leads to a chain of logical consequences that contradict the limited edge density of these graphs. This automatically implies the existence of a sufficiently small negative eigenvalue, and hence a deep minimum of the cosine sum.
In their published paper, the authors proved that for any set of N integers, the corresponding cosine sum necessarily takes a value below -N^(1/10). For small N, this estimate appears modest, but at extreme scales, it becomes fundamentally significant. For N = 10²⁰, it yields a minimum below -100, which is orders of magnitude better than Ruzsa's old estimate.
Just two days after its publication, an independent result by Benjamin Bedert appeared, obtained using classical Fourier analysis. His bound turned out to be slightly stronger: the minimum is below -N^(1/7). For N = 10²⁰, this corresponds to a value of about -720.
The primary significance of these works lies not only in the numerical estimates. For the first time in decades, rigorous results took a power-law form in N, as in Chowla's original conjecture, where the bound was -N^(1/2). Previously known estimates did not have this structure.
Although a complete proof of Chowla's conjecture is still a long way off, the new connection between graph theory and Fourier analysis has transformed the problem space itself. For the first time, different fields of mathematics converged in a single logical construct, demonstrating that problems of spectral analysis, graph structure, and classical Fourier series can describe the same fundamental phenomena from different perspectives.
Two hundred years ago, Joseph Fourier proposed an idea that would eventually become the foundation of modern mathematics and physics. He proposed that almost any function can be represented as a sum of simple waves. Today, this principle, which underlies the Fourier transform , is used in analyzing the spectra of distant stars, processing signals, investigating the structure of matter, and studying processes occurring deep beneath the Earth's crust . Despite the universality of the method, questions remain surrounding it that have defied rigorous mathematical analysis for decades.
One such problem was formulated in 1965 by mathematician Sarvadaman Chowla. He became interested in an extremely simple but fundamental object: the sum of cosines without coefficients, where each function has the same amplitude and differs only in frequency. Formally, this is an expansion of the form cos(a₁x) + cos(a₂x) + ... + cos(aₙx), where a₁, a₂, …, aₙ are integers. From the perspective of Fourier theory, this is one of the most elementary types of series, but it turned out to be unexpectedly difficult to analyze.
The maximum value of such a sum is determined trivially. At x = 0, each cos(ax) function is equal to 1, so the sum of N cosines always reaches the value N. The minimum, by contrast, is much more complex. The minimum points of individual waves do not coincide in time, and the behavior of the sum is determined by complex interference of different frequencies. A fundamental question arises: how deeply can the graph of such a sum descend into negative territory?
Chowle knew of examples of sets of N integers for which the minimum sum approached a value of the order of -√N. He also observed that most other sets yielded even more negative values. This led him to the conjecture that for any set of N positive integers, the corresponding cosine sum necessarily takes a value below -√N, as well as to the more precise question of how quickly this lower bound decreases as N increases.
Despite the simplicity of its formulation, the problem remained virtually unsolved for decades. In 2004, Imre Ruzsa obtained a result that was long considered the best known. His estimate provided an extremely weak lower bound on large scales. For example, for a sum of 10²⁰ cosines, it guaranteed a value below approximately -7, while Chowla's conjecture suggested a level of about -10¹⁰. For nearly 20 years, this estimate remained the pinnacle of progress.
The shift occurred unexpectedly and came from a completely different area of mathematics. Four researchers—Zhihan Jin, Aleksa Milojevic, Istvan Tomon, and Shengtong Zhang—were studying graph theory problems, specifically the MaxCut problem. It involves optimally partitioning a graph into two parts so as to maximize the number of edges between them. This problem has both theoretical significance and applied interpretations in physics, computer science, and engineering, but is NP-hard and lacks a universal algorithmic solution.
In their research, the team analyzed the spectral characteristics of graphs, that is, the eigenvalues of the matrices describing their structure. These quantities reflect fundamental properties such as connection density, connectivity, and the presence of clusters. Particular attention was paid to negative eigenvalues, as they were found to be closely related to MaxCut scores. The study yielded a general result: if a graph lacks sufficiently small negative eigenvalues, its structure is inevitably dominated by cliques—dense subgraphs where every vertex is connected to every other.
A further turnaround occurred after a letter from Ilya Shkredov, who pointed out the connection between Chowla's problem and a special class of objects—Cayley graphs. These graphs are constructed over a set of integers and a prime modulus, and their spectral characteristics are directly related to the values of the corresponding cosine sums. The minimum eigenvalue of such a graph coincides with the smallest value of the cosine sum.
Using previously obtained results on MaxCut, the researchers were able to reformulate the problem. Instead of a direct analysis of the eigenvalues, it was now sufficient to prove that large cliques cannot exist in the corresponding Cayley graphs. The assumption of their existence leads to a chain of logical consequences that contradict the limited edge density of these graphs. This automatically implies the existence of a sufficiently small negative eigenvalue, and hence a deep minimum of the cosine sum.
In their published paper, the authors proved that for any set of N integers, the corresponding cosine sum necessarily takes a value below -N^(1/10). For small N, this estimate appears modest, but at extreme scales, it becomes fundamentally significant. For N = 10²⁰, it yields a minimum below -100, which is orders of magnitude better than Ruzsa's old estimate.
Just two days after its publication, an independent result by Benjamin Bedert appeared, obtained using classical Fourier analysis. His bound turned out to be slightly stronger: the minimum is below -N^(1/7). For N = 10²⁰, this corresponds to a value of about -720.
The primary significance of these works lies not only in the numerical estimates. For the first time in decades, rigorous results took a power-law form in N, as in Chowla's original conjecture, where the bound was -N^(1/2). Previously known estimates did not have this structure.
Although a complete proof of Chowla's conjecture is still a long way off, the new connection between graph theory and Fourier analysis has transformed the problem space itself. For the first time, different fields of mathematics converged in a single logical construct, demonstrating that problems of spectral analysis, graph structure, and classical Fourier series can describe the same fundamental phenomena from different perspectives.
