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<\/p>\nWith his assumptions in place, Cohen moved on to the actual proof. He quickly realized that this was unlike any Fourier-related problem he had worked on before. \u201cI tried using all my tools to prove the fractal uncertainty principle, and none of them even came remotely close to working,\u201d he said.<\/p>\n
Feeling stuck, he went back to Dyatlov and Bourgain\u2019s proof of the principle in one dimension and sought to understand exactly how it worked.<\/p>\n
Dyatlov and Bourgain used an uncommon method in their proof. It involved isolating one peak from a fractal-like function at a time and showing that the Fourier transform of that peak would spread out. Doing this for all peaks, and considering how the Fourier transforms would add together, they proved that the total Fourier transform could never equal zero often enough to form a fractal \u2014 there wouldn\u2019t be enough holes.<\/p>\n
<\/p>\n
Jean Bourgain, seen here in 2012, was an author on more than 500 papers spanning much of mathematics.<\/p>\n
Isolating each peak required constructing a very specific function that, when multiplied by the original fractal-like function, would pull out just the peak and be close to zero everywhere else. This is called a damping function, and it needs to be perfectly tailor-made to work. \u201cThis is a challenging thing to construct,\u201d Cohen said. But he knew that if he could do it in higher dimensions, he could unlock the entire proof.<\/p>\n
Cohen consulted Dyatlov about his plan to construct this special function. Before Bourgain died in late 2018, he too struggled with this problem, and he shared his unpublished notes with Dyatlov. Now, Dyatlov shared them with Cohen. \u201cBourgain was a legendary analyst,\u201d Cohen said. Reading the note felt like \u201creceiving this unfinished knowledge from him.\u201d<\/p>\n
The notes contained exactly the hint Cohen needed. \u201cIt just blew my mind,\u201d Cohen said. \u201cIt really unlocked the problem for me.\u201d<\/p>\n
Before reading Bourgain\u2019s note, Cohen had a few ideas for how to construct the damping function, but they were highly complicated and precise, like the designs for building a house brick by brick. The note revealed an unexpected way to do it. It involved taking a detour into complex analysis \u2014 the study of functions of imaginary numbers, which include the square root of negative 1. This detour allowed Cohen to build a much more flexible object, which could then be used to construct the damping function indirectly.<\/p>\n
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Semyon Dyatlov, a mathematician at the Massachusetts Institute of Technology, proved the one-dimensional fractal uncertainty principle in 2016.<\/p>\n
Armed with this insight, Cohen then needed to find a way to create just the right version of this flexible object to produce a proper damping function. \u201cTo construct something like this that has very specific properties is highly nontrivial. It\u2019s delicate,\u201d said Wilhelm Schlag<\/a> of Yale University, with whom Cohen studied as an undergraduate. \u201cIn two dimensions, nobody knew how to do that, and Alex came up with a brilliant construction of such a thing.\u201d<\/p>\n