A new paper posted this week on the arXiv this week by UW grad students Evan Dummit and Márton Hablicsek answers a question left open in a paper of mine with Richard Oberlin and Terry Tao. Let me explain why I was interested in this question and why I like Evan and Marci’s answer so much!

Recall: a **Kakeya set** in an n-dimensional vector space over a field k is a set containing a line (or, in the case k = R, a unit line segment) in every direction. The “Kakeya problem,” phrased loosely, is to prove that Kakeya sets cannot be too small.

But what does “small” mean? You might want it to mean “measure 0” but for the small but important fact that in this interpretation the problem has a negative answer: as Besicovitch discovered in 1919, there are Kakeya sets in R^2 with measure 0! So Kakeya’s conjecture concerns a stronger notion of “small” — he conjectures that a Kakeya set in R^n cannot have Hausdorff or Minkowski dimension strictly smaller than n.

(At this point, if you haven’t thought about the Kakeya conjecture before, you might want to read Terry’s long expository post about the Kakeya conjecture and Dvir’s theorem; I cannot do it any better here.)

The big recent news in this area, of course, is Dvir’s theorem that that the Kakeya conjecture is true when k is a finite field.

Of course one hopes that Dvir’s argument will give some ideas for an attack on the original problem in R^n. And that hasn’t happened yet; though the “polynomial method,” as the main idea of Dvir’s theorem is now called, has found lots of applications to *other* problems in real combinatorial geometry (e.g. Guth and Katz’s proof of the joints conjecture.)

Why not Kakeya? Well, here’s one clue. Dvir actually proves *more* than the Kakeya conjecture! He proves that a Kakeya set in F_q^n *has positive measure.*

(Note: F_q^n is a finite set, so of course *any* nonempty subset has positive measure; so “positive measure” here is shorthand for “there’s a lower bound for the measure which is bounded away from 0 as q grows with n fixed.”)

What this tells you is that R really is different from F_q with respect to this problem; if Dvir’s proof “worked” over R, it would prove that a Kakeya set in R^n had positive measure, which is false.

So what’s the difference between R and F_q? In my view, it’s that R has multiple scales, while F_q only has one. Two elements in F_q are either the same or distinct, but there is nothing else going on metrically, while distinct real lines can be very close together or very far apart. The interaction between distances at different scales is your constant companion when working on these problems in the real setting; so maybe it’s not so shocking that a one-scale field like F_q is not a perfect model for the phenomena we’re trying to study.

Which leads us to the ring F_q[[t]] — the “non-archimedean local ring” which Dummit and Hablicsek write about. This ring is somehow “in between” finite fields and real numbers. On the one hand, it is “profinite,” which is to say it is approximated by a sequence of larger and larger finite rings F_q[[t]]/t^k. On the other hand, it has infinitely many scales, like R. From the point of view of Kakeya sets, is it more like a finite field, or more like the real numbers? In particular, does it have Kakeya sets of measure 0, making it potentially a good model for the real Kakeya problem?

This is the question Richard, Terry, and I asked, and Evan and Marci show that the answer is **yes**; they construct explicitly a Kakeya set in F_q[[t]]^2 with measure 0.

Now when we asked this question in our paper, I thought maybe you could do this by imitating Besicovitch’s argument in a straightforward way. I did not succeed in doing this. Evan and Marci tried too, and they told me that this just plain doesn’t work. The construction they came up with is (at least as far as I can see) completely different from anything that makes sense over R. And the way they prove measure 0 is extremely charming; they define a Markov process such for which the complement of their Kakeya set is the set of points that eventually hit 0, and then show by standard methods that their Markov process goes to 0 with probability 1!

Of course you ask: does their Kakeya set have Minkowski dimension 2? Yep — and indeed, they prove that any Kakeya set in F_q[[t]]^2 has Minkowski dimension 2, thus proving the Kakeya conjecture in this setting, up to the distinction between Hausdorff and Minkowski dimension. (Experts should feel free to weigh in an tell me how much we should worry about this distinction.) Note that dimension 2 is special: the Kakeya conjecture in R^2 is known as well. For every n > 2 we’re in the dark, over F_q[[t]] as well as over R.

To sum up: what Dummit and Hablicsek prove makes me feel like the Kakeya problem over F_q[[t]] is, at least potentially, a pretty good model for the Kakeya problem over R! Not that we know how to solve the Kakeya problem over F_q[[t]]…..

Does their argument work for finite extensions of Q_p ?

Good question! Their lower bound for Minkowksi dimension works, but they don’t construct a Kakeya set of measure 0 in that setting. Someone should!

A nice result!

The difference between Hausdorff and (lower) Minkowski dimension is that to compute Minkowski dimension, one covers a set by balls of constant radius, whereas to compute Hausdorff dimension, one has to also consider covers by balls of variable radius. As such, proving lower bounds for Hausdorff dimension is a bit harder than Minkowski. For Kakeya, though, there is a pigeonholing trick due to Bourgain which almost allows one to reduce the Hausdorff problem to the Minkowski one, but at a light cost: instead of covering a Kakeya set (that contains a line in each direction) by delta-balls, one covers a slightly smaller set by delta balls, one which contains 1/log(1/delta) of a line in 1/log(1/delta) of the directions. (I’m fudging a bit with the log factors, but this is roughly right.) In practice, these log factors don’t change the powers of delta which are the important thing in dimension bounds, so Hausdorff and (lower) Minkowski bounds for Kakeya are “morally” equivalent, though in practice it can be a pain to adapt the Minkowski argument to the Hausdorff setting.

Interesting. What if for some fixed d we require a d-dimensional disc in every direction (or, over a finite field, a d-dimensional affine subspace in any dimension)? Is there an easy reduction (or at least a known reduction) to d=1?

Over a finite field, this forces the set to have measure 1: more precisely, in Prop 4.16 of the Ellenberg-Oberlin-Tao paper we prove that a subset of F_q^n containing a d-plane in every direction has size at least

q^n(1 – q^{1-d})^(n choose 2)

at least when q is sufficiently large relative to n.

As for the Euclidean setting: per the Wikipedia article on Kakeya sets, every such set is expected to have positive measure; the best result is due to Bourgain, giving an affirmative answer when d is at least on order of log n.

[…] About a year ago I wrote about the handsome theorem of Dummit and Hablicsek that there exist Besicovich sets over F_q[[t]]; that is, there are sets of measure 0 which contain a line in every direction. As I explained in that post, power series rings over finite fields are promising intermediate contexts for problems in additive combinatorics; like the real numbers, they have infinitely many scales, but unlike the real numbers, those scales naturally form a discrete set, and the whole ring decomposes in some iterative sense into a bunch of copies of a finite field, where things are much simpler (though by no means simple!) […]

[…] with applications to Kakeya sets and mergers, by Dvir, Kopparty, Saraf and Sudan. – Kakeya sets over non-archimedian lcoal rings – Variants of the Kakeya problem over an algebraically closed field by Kaloyan Slavov. […]

[…] that aren’t fields; extending our knowledge about additive combinatorics to such settings is a long-standing interest of mine. Second, the polynomial method over finite fields usually works in the “fixed dimension […]

[…] One reason they’re different is that R has scales and F_p does not. Two distinct real numbers can be very close together or very far apart. In a finite field, two numbers are either the same or they’re not. A better analogue for R (of course I’d think this, I’m a number theorist) is the ring of p-adic integers, Z_p, which metrically looks a lot more like the reals. Even the finite ring Z/p^k Z has scales, though only k, not infinitely many. (For more about this, see this old post.) […]