Merging Fields, Mathematicians Go the Distance on Old Problem

In the School of Mathematics, AMIAS Member Rachel Greenfeld and past Veblen Research Instructor Sarah Peluse (2020–23), with their collaborator Marina Iliopoulou, have proved that large sets of points with integer distances between them must lie almost entirely on a single line or circle. 

"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?"

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