Original Title: “On the polynomial Wolff axioms”
Authors: Nets H. Katz (California Institute of Technology) and Keith M. Rogers (ICMAT)
Source: Geometric and Functional Analysis, 28 (6), pp 1706-1716
Date of online publication: September 14th, 2018
Link: https://link.springer.com/article/10.1007/s00039-018-0466-7
Fourier analysis describes the process of decomposing a signal into its frequencies and then recomposing the signal from the frequencies. It has proved to be an extremely useful tool in mathematics, physics and technology. For example, for storing and sending sound files in a more efficient way. The human ear cannot hear very high or very low frequencies, so these can be discarded before storing the Fourier transform of a signal, thus saving memory. In order to reproduce the new sound, the signal is recomposed by adding up the terms of the series.
Unfortunately, however, the process is not infallible, and it is not always true that when the frequencies are summed up again, the resultant signal sounds like the original one. Whether Fourier series, in space R^n with n>1, converge to the original signal or not is connected to whether tubes with arbitrary positions, but different directions, overlap a lot or not. This is because the signal can be further decomposed into wave-packets which essentially live on the tubes. Although there may be cancelation when summing up the wave-packets, the main issue, ignoring the oscillation, is how much the tubes can intersect. This is quantified precisely in the Kakeya conjecture.
The first connection between Fourier series and the Kakeya conjecture was made in a seminal work by Charles Fefferman [6] in 1971. This connection was later fleshed out in works by Jean Bourgain [1] in 1991 and Terence Tao [13] in 1999. However, the Kakeya conjecture had been studied much earlier, for other reasons. The problem, as initially considered, was to determine how large a set must be for a needle to be continuously turned around within it, pointing in every possible direction as it turns, before eventually returning to its original position. Surprisingly, sets exist that satisfy this property and which have arbitrarily small Lebesgue measure (the technical term for area or volume). Indeed, there are Kakeya sets (sets containing a unit line segment in every direction) with zero measure.
However, there are zero measure sets which are smaller than others. For example, a line in R^n is clearly smaller than a plane because it is one-dimensional, whereas the plane is two-dimensional. The Kakeya conjecture states that Kakeya sets must have dimension n. That is to say, although they can have zero measure, they cannot be any smaller than that.
The Kakeya conjecture in R^2 was solved by Antonio Córdoba [3] and Roy Davies [4] in the 1970s (the oscillatory versions of the problem were solved by Charles Fefferman [5], Lennart Carleson and Per Sjölin [2]). However, for higher dimensions, the conjecture has resisted the best efforts of the harmonic analysis community ever since.
Now, in [11], Nets Katz and Keith Rogers have proved a weak form of the Kakeya conjecture in higher dimensions, under the assumption that the line segments have some additional algebraic structure. The problem is first discretized, so that the angle between any two-line segments is greater than 1/x, where x is a large number. If we assume that the line segments are all found on the surface of a two-dimensional cone in R^3, it is easy to deduce that there can be no more than a constant multiple of x line segments. In [11], the correct bound was proved for any real algebraic variety in any dimension, confirming a conjecture of Larry Guth [7]. Furthermore, a generalized version was proved that considers semi-algebraic sets rather than algebraic varieties, thereby solving a problem posed by Guth and Joshua Zahl [8].
Furthermore, in [9], Jonathan Hickman and Rogers proved that if a Kakeya set has no algebraic structure at all, then in that case the Kakeya conjecture is also true. Balancing between this and the result of [11], they showed that Kakeya sets cannot be too small in any case, even when the sets have an intermediate amount of algebraic structure. For certain ambient dimensions n, this argument improves the best previously known lower bounds (obtained by Katz and Tao [12], on the one hand, and by Wolff [14] on the other) for the fractal dimension of any Kakeya set. Progress on one of the oscillatory versions of the problem was also obtained in [10].
