A deceptively simple math proposition known as the Kakeya conjecture underpins a tower of other questions in physics, number theory, and harmonic analysis.
Here’s one way to think about the Minkowski dimension: Take your set and cover it with tiny balls that each have a diameter of one-millionth of your preferred unit. If your set is a line segment of length 1, you’ll need at least 1 million balls to cover it. If your set is a square of area 1, you’ll need many, many more: a million squared, or a trillion. For a sphere of volume 1, it’s about 1 million cubed , and so on. The Minkowski dimension is the value of this exponent.
These dimensions are familiar. But using Minkowski’s definition, it becomes possible to construct a set that has a dimension of, say, 2.7. Though such a set doesn’t fill up three-dimensional space, it’s in some sense “bigger” than a two-dimensional surface. When you cover a set with balls of a given diameter, you’re approximating the volume of the fattened-up version of the set. The more slowly the volume of the set decreases with the size of your needle, the more balls you need to cover it. You can therefore rewrite Davies’ result—which states that the area of a Kakeya set in the plane decreases slowly—to show that the set must have a Minkowski dimension of 2.
Mathematicians also want to know whether they can rebuild the original function if they’re given just some of its infinitely many constituent frequencies. They have a good understanding of how to do this in one dimension. But in higher dimensions, they can make different choices about which frequencies to use and which to ignore. Fefferman proved, to his colleagues’ surprise, that you might fail to rebuild your function when relying on a particularly well-known way of choosing frequencies.
His proof hinged on constructing a function by modifying Besicovitch’s Kakeya set. This later inspired mathematicians to develop a hierarchy of conjectures about the higher-dimensional behavior of the Fourier transform. Today, the hierarchy even includes conjectures about the behavior of important partial differential equations in physics, like the Schrödinger equation. Each conjecture in the hierarchy automatically implies the one below it.
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