New paper on the arXiv by me, Richard Oberlin, and Terry Tao: “Kakeya set and maximal conjectures for algebraic varieties over finite fields.” The paper got started in an interesting way. I read about Dvir’s proof of the finite field Kakeya set conjecture on Terry’s blog. To an algebraic geometer the proof is extremely striking — it uses so little algebraic geometry that one thinks (encouraged by one’s natural prejudices) “surely a little more algebraic geometry will lead to even better results!” I had some back and forth with Terry in the comments on his post, then posted a little more about the problem on this blog, learning much more about the problem from the comments I received from Terry and others.
That got me, Terry, and Richard started thinking about whether the polynomial method could be used to get the Kakeya maximal conjecture over finite fields. It’s something rather distant from the usual problems I work on (apart from my periodic and so far fruitless obsession with the cap set problem.) The paper simply wouldn’t exist if it weren’t for blogging and the opportunities it provides for fast, informal idea-sharing between multiple mathematicians in different specialties and physical locations.
Terry has ably described the main thrust of the paper over at his place; the idea is that we use the “polynomial method” of Dvir, together with some methods from harmonic analysis, to prove the finite-field analogue of the Kakeya maximal conjecture — very imprecisely, this says that if f is a function on F_p^n with small norm and L is a set of lines such that f|l has large norm for every l in L, then L itself can’t cover too many different directions. Of course, L can be big — a function which is big on a single hyperplane is big on tons of distinct lines.
As I said, Terry explained that part on his blog, so let me say a little something about the extension from functions on F_p^n to functions on an arbitrary variety over F_p. There’s some content here even if you only care about functions on vector spaces!
For instance: Dvir’s theorem says that a subset E of F_p^2 containing a line in every direction must have at least cp^2 points. Essentially the same argument tells you that if E contains a conic with an asymptote in every direction, then again |E| > cp^2. From the result in our paper, on the other hand, one can show easily that if E contains a conic with an asymptote along every horizontal line, then again |E| > cp^2.
Why is the last problem along the same lines as the first two? In the second problems, we are asking E to contain a conic in every asymptotic direction; that is, E contains (the affine F_p-points of) a set of conics C whose union covers the F_p-points of the line at infinity.
But there’s nothing special about P^2 and the line at infinity; in fact, as our result shows, the same kind of theorem holds with P^2 replaced by an arbitrary variety X and the line at infinity with an arbitrary hypersurface W in X. For instance, one might take X to be P^2 blown up at the point [1:0:0], and W the exceptional divisor; then a set of conics whose union covers the F_p-points of W is precisely a set C of conics in the affine plane such that every horizontal line is an an asymptote for some conic in C.