The Kakeya problem, in its best known formulation, is the following. Let be a set which contains a translate of every unit line segment; equivalently, for every direction , contains a unit line segment parallel to . An -dimensional ball of radius 1/2 is a simple example of a set with this property, but there are many other such sets, some of which have -dimensional measure 0 (an old construction of Besicovitch). Can be even smaller than that and have Hausdorff dimension strictly smaller than ? The conjectured answer is no. This is known in dimension 2 (due to Davies), but open otherwise.
This is one of the “big” problems in harmonic analysis, with an interesting history, relations to several major conjectures in the field, and intriguing connections to other areas of mathematics. There are many expositions available; here is the link to my latest one, in case you are interested.
There is a variant of the problem set in a vector space over a finite field, rather than in the Euclidean space. Let be a large prime, and let be the -dimensional vector space over the -element field . A line in is the -element set , where are fixed vectors and . Note that if is a multiple of , then the lines defined by and coincide.
A Kakeya set is a set which contains a translate of every line. Is it true that every Kakeya set must have cardinality at least , where is a constant depending only on the dimension , but not on ? This question was first formulated in an expository paper by Tom Wolff in the late 1990s, and addressed in several subsequent papers (Mockenhaupt-Tao 2004, Tao 2005, Bourgain-Katz-Tao 2004).
The reason for this post is that this problem appears to have just been solved! Only a few days ago, Zeev Dvir posted a paper on the arXiv where he proves that Kakeya sets in must have cardinality at least for any , with the constant independent of . This is best possible except for the endpoint.
The idea of the proof is quite beautiful and genuinely new in this context. Consider the space of all homogeneous polynomials over of degree in variables , where ; this space has dimension . If the Kakeya set has cardinality smaller than that, then there is a polynomial, say , which vanishes on . But because is Kakeya set, this implies via some simple algebraic manipulations that must in fact have at least zeros, which contradicts a theorem of Schwartz and Zippel bounding the number of such zeros. A well-known product set argument completes the proof of the bound stated above.
There is, of course, a verification process that has to take place. However, Dvir’s argument is very short (only about two pages) and easy to check. I’m not familiar with the Schwartz-Zippel theorem, which plays a key role in the proof, but I’m sure that it is receiving a lot of renewed scrutiny right now.
Is this how the “real” Euclidean problem will be solved? First of all, I don’t expect any 2-page proofs in the Euclidean case. There are just too many technicalities to deal with. That said, Dvir’s idea could be the most significant new contribution to the subject in years. It has been suspected, now and then, that further improvements in Kakeya bounds might involve some sort of algebraic structure – except that of course we didn’t know how to do it. While Dvir’s argument is specific to finite fields, there is a very good chance that it will inspire further work in the Euclidean setting. I have no idea whether this might actually resolve the problem, but I’m pretty sure that we should at least see further developments in this direction.
Update, March 25: Dvir’s paper has now been updated to include an additional argument (due to Alon and Tao) that proves the endpoint estimate. This resolves completely the finite field Kakeya conjecture.