Mathematicians are still trying to understand fundamental properties of the Fourier transform, one of their most ubiquitous and powerful tools. A new result marks an exciting advance toward that goal.

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.

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