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If a large (but non-infinite) set of points are whole-number distances away from each other, how can they be arranged? A new result proves that a circle is one of the only options.

The change of plans came on a road trip. On a beautiful day last April, the mathematicians Rachel Greenfeld and Sarah Peluse set out from their home institution, the Institute for Advanced Study in Princeton, New Jersey, heading to Rochester, New York, where both were scheduled to give talks the next day.

They had been struggling for nearly two years with an important conjecture in harmonic analysis, the field that studies how to break complex signals apart into their component frequencies. Together with a third collaborator, Marina Iliopoulou, they were studying a version of the problem in which the component frequencies are represented as points in a plane whose distances from each other are related to integers. The three researchers were trying to show that there couldn’t be too many of these points, but so far, all their techniques had come up short.

They seemed to be spinning their wheels. Then Peluse had a thought: What if they ditched the harmonic analysis problem — temporarily, of course — and turned their attention to sets of points in which the distance between any two points is exactly an integer? What possible structures can such sets have? Mathematicians have been trying to understand integer distance sets since ancient times. For example, Pythagorean triples (such as 3, 4 and 5), represent right triangles whose three vertices are all integer distances apart.

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