# The Kakeya problem, and connections to harmonic analysis

What is a Kakeya set?

Questions in harmonic analysis

The Kakeya conjecture

This page is meant to provide a very brief introduction to the Kakeya problem and some related questions. It was first posted in January 2001, and I expect that it will be evolving over time. For one thing, I will update it if further progress is made on the Kakeya conjecture. Also, I may include: descriptions of other problems, either related directly to Kakeya or just similar in spirit; more details about some of the ideas and methods involved; additional links, references, etc. If you have any comments or suggestions regarding this page, please write to ilaba@math.ubc.ca.

### What is a Kakeya set?

A Besicovitch set is a subset of Rn which contains a unit line segment in each direction. Besicovitch sets are also known as Kakeya sets.

Besicovitch sets have an interesting history. In 1917 Besicovitch was working on a problem in Riemann integration, and reduced it to the question of existence of planar sets of measure 0 which contain a line segment in each direction. He then constructed such a set, and published his construction in a Russian journal in 1920.

Due to the civil war and the blockade, there was hardly any communication between Russia and the rest of the world at the time. (I was rather surprised to hear that anybody could be bothered to publish mathematical journals in Russia in 1920! But this is what Besicovitch says in Amer. Math. Monthly 70 (1963), 697-706.) Thus Besicovitch did not know that a Japanese mathematician Kakeya had asked, also in 1917, a somewhat related question: what is the smallest area of a convex set within which one can rotate a needle by 180 degrees in the plane? Pal (1921) resolved that problem (the convex set should be an equilateral triangle); the more interesting question, without the convexity assumption, remained open. Besicovitch was told of all this several years later, after he left Russia. By modifying his original construction, he gave the surprising answer that the area in question must of course be positive, but (with a lot of patience) may be made arbitrarily small. His solution was published in 1928.

Many other ways to construct Besicovitch sets of measure zero have since been discovered (Perron, Schoenberg, Kahane,...). Most of them rely on the idea illustrated below. Essentially, we slice a triangle into many thin subtriangles, and then rearrange the latter so that they overlap a lot. If one does this carefully and then takes the limit as the number of subdivisions goes to infinity, one gets a Besicovitch set of measure 0 in the plane. Straightforward product techniques extend the construction to higher dimensions.

### Questions in harmonic analysis

Kakeya sets (as well as a closely related construction due to Nikodym, 1927) have long been used to construct various counterexamples in analysis, starting with that Riemann integration problem considered by Besicovitch in 1917. Here I'm only going to describe two central problems in harmonic analysis - the restriction and Bochner-Riesz conjectures - which seem to depend on just how small Kakeya sets can really be. There are many other open problems in analysis, as well as PDE and number theory, involving Kakeya sets. (See the references in the "Further reading" section for more info.)

1. Convergence of Fourier series. In 1971, C. Fefferman used Besicovitch sets to construct a counterexample to the ball multiplier conjecture, which concerned a very basic problem in harmonic analysis. Let f be a function in Lp(Rn), and let F be its Fourier transform. If SR f ( x ) is the integral of F(k) exp(2 \pi i k x) over |k| < R, do SR f converge to f in Lp as R goes to infinity? It has long been known that Lp convergence holds for p = 2, and fails for p = 1 or infinity, in all dimensions. An old theorem of M. Riesz says that in dimension 1 the answer is yes for all finite p > 1, and a higher-dimensional analogue was generally expected to be true. However, Fefferman proved that in dimensions > 1 convergence fails for all p other than 2!

Fefferman's argument showcases a beautiful interplay between multidimensional Fourier analysis and Euclidean geometry. Here is how it goes. The problem is equivalent to checking whether the disc multiplier operator, which we will call S, is bounded on Lp(Rn). It now suffices to find functions f in Lp(Rn) such that the Lp norms of Sf are large. Fefferman does this by letting f be a sum of characteristic functions of long and thin tubes, multiplied by appropriate phase factors. It turns out that S essentially shifts each tube by a fixed distance in the "long" direction. Now suppose that the shifted tubes form something close to a Kakeya set; then the support of Sf is small, and so by Holder's inequality its Lp norm is large.

Now the disc multiplier question can be rewritten as follows. Let f and F be as above, and let g(k) be the characteristic funtion of the unit ball; does the inverse Fourier transform of F(k) g (k/R) converge to f in Lp(Rn) as R goes to infinity? Fefferman's result says that it is always so only for p=2. Yet if we let g to be e.g., a Gaussian instead, then the answer is yes for all finite p > 1 - this is the standard textbook proof of the Fourier inversion formula. So what happens if g(k) is somewhere in between? Where exactly does convergence start to fail?

The Bochner-Riesz conjecture, which looks essentially like a "regularized" version of the disc multiplier conjecture, deals with exactly this question. It is generally regarded as one of the major unsolved problems of harmonic analysis. (See Stein's book for a comprehensive discussion.) It turns out that Fefferman's construction would provide a counterexample to the Bochner-Riesz conjecture, if one could construct a Besicovitch set in Rn of Hausdorff dimension less than n. We will have much more to say about the latter question later on.

2. The restriction problem. Let f be a function in Lp(Rn), can we say anything intelligent about the restriction of the Fourier transform of f to a lower-dimensional subset E of Rn? If E is a hyperplane, then we can't - the Fourier transform of f does not even have to be defined on E. But if E is a curved surface, for example a sphere, then things are a bit different. There are deep results, due to Stein, Tomas, Fefferman, Bourgain, Wolff, and others, which say that for certain values of p and q the Fourier transform of f is actually in Lq(E). However, what has been proven is still quite far from what is being conjectured.

Again, the n-dimensional geometry plays a major role, and there is a point at which Kakeya sets become crucial (this time, via the uncertainty principle). Here is, very roughly, why. Let E=Sn-1 be the unit sphere. By duality, the restriction problem is equivalent to estimating the Lq' norm of the Fourier transform of measures supported on the sphere. Let m be such a measure concentrated on a very small spherical cap of diameter r. By uncertainty principle, its Fourier transform is essentially constant on a tube of length r-2 and diameter r-1, pointing in the direction perpendicular to the cap. Multiplying our measure m by a suitable phase factor, we can place this tube anywhere we like in the Fourier space.

Now consider the sum of a large number of such measures, multiplied by phase factors chosen so that the corresponding Fourier tubes form an approximate Kakeya set. As in the disc multiplier argument, the Fourier transform of the whole thing has small support and therefore large Lp norms.

The restriction and Bochner-Riesz problems have long been known to be connected (e.g., partial results on both problems would often follow from the same oscillatory integral estimate). One can trace relevant research back to the 1930's, perhaps further. Many significant contributions are due to Stein, Calderon, Zygmund, Carleson, Sjolin, Fefferman, Hormander, Tomas, Cordoba, Christ, Sogge, Carbery, Bourgain, Wolff, Moyua, Tao, Vargas, Vega,... And it doesn't look like we'll be done with it any time soon.

### The Kakeya conjecture

The problem below looks like geometric measure theory. The motivation for studying it comes from harmonic analysis, analytic number theory, and PDE. And the techniques used to prove the partial results stated below are mostly geometrical and combinatorial, additive number theory being the latest addition. It is generally expected that ideas from other, seemingly unrelated, fields of mathematics will be needed to finally resolve the problem. Anyone looking for opportunities for interdisciplinary research?

Conjecture. A Besicovitch set in Rn must have dimension n.

Naturally, one can ask what I mean by "dimension". There are several, not quite equivalent, definitions - those of interest to us are the Hausdorff and Minkowski dimension. Most mathematicians know what the Hausdorff dimension is. The Minkowski dimension (of a compact set) may be defined as follows. Let Eh be the h-neighbourhood of E in Rn, then the (upper) Minkowski dimension of E is the infimum of all a such that |Eh| < C hd-a for some constant C. The upper Minkowski dimension of a set is always greater or equal to its Hausdorff dimension, and there are examples of sets for which the inequality is strict.

There is also a stronger formulation of the conjecture in terms of maximal functions. The maximal function statements are actually quite important, as they are very closely related to the problems in analysis mentioned above. This is a bit more technical, though, so we will not go into it here - at least for now.

The conjecture is known to be true in dimension 2: the Hausdorff (hence also Minkowski) version was proved by Davies in 1971, and the maximal function version is due to Cordoba (1977) and Bourgain (1991). In higher dimensions it is still far from settled. Here is a brief summary of the best currently known lower bounds. Since there are no "less-or-equal" and "greater-or-equal" symbols in HTML, I will use <= and >=.
• The Minkowski dimension of a Besicovitch set in Rn is at least 5/2 + 10-10 for n=3 (Katz-Laba-Tao 1999), 3 + 10-10 for n=4 (Laba-Tao 2000), (2 - 21/2)(n-4)+3 for 4 < n < 24 (Katz-Tao 2001), and (n + t -1)/t for n >= 24, where t=1.67513... is the root of t3 - 4 t + 2 = 0 that lies between 1 and 2 (Katz-Tao 2001).

• The Hausdorff dimension of a Besicovitch set in Rn is at least (n + 2 ) / 2 for n = 3,4 (Wolff 1994), and at least (2 - 21/2)(n-4)+3 for n > 4 (Katz-Tao 2001).
Here are some of the ideas that went into proving the above results.
• The bush argument, due to Bourgain. Suppose that the set E contains a unit line segment in each direction. In order for E to have low dimension, the segments must intersect a lot. Hence there are many "high multiplicity" points. Let P be such a point, then there are many line segments through P; since all of them go through P, they must be disjoint otherwise. An appropriate quantitative version of this argument produces a lower bound on the dimension of E.

I was going to include a picture, but then I found this on the Web. It's much better than anything I could come up with.

• Wolff's hairbrush argument is similar in spirit to the "bush" argument, but one must do the combinatorics a lot more carefully. Instead of using just one point of high multiplicity, one finds a line segment L consisting largely of such points, and computes the volume of the "hairbrush" of all line segments intersecting L.

Looking at the additive number theory approach (see below), one might be tempted to regard the hairbrushes as obsolete. But for the time being, this argument (combined with the additional geometrical analysis of Laba-Tao) is still the best available in dimensions 4 to 8.

• Additive number theory arguments, again first used in this context by Bourgain. (NB: Bourgain's results are not included in the above list of "world records", as they have been improved upon by others, but he has contributed more than anyone else towards understanding Kakeya and its relationship to harmonic analysis.)

Up until now, the philosophy has been to prove that high multiplicity leads to the occurrence of things like a bush and hairbrush, consisting of many essentially disjoint line segments and therefore having relatively large size. In a way, these are "local" arguments. This is about to change. The new idea is to prove that if high multiplicity persists throughout our Kakeya set, it actually forces the set to be very regular and symmetric - to the extent that many of the line segments must be parallel. Which of course contradicts the above definition of Kakeya sets.

To make this kind of reasoning rigorous, one uses something called inverse additive number theory. Bourgain's argument (improved later by Katz-Tao) draws on the recent work of Gowers on Szemeredi's theorem - of particular interest is Gowers's take on the Balog-Szemeredi theorem, which roughly speaking says that sets with a small difference set have "structure". For more details, see the references below.

At first, it didn't seem likely that these techniques could improve on the hairbrush in low dimensions, in particular in dimension 3. But Katz-Laba-Tao used geometrical arguments (including the hairbrush) to "factor out" one dimension and obtain a setting in which Bourgain's argument is again effective.

There are several recent expository articles on Kakeya and related topics:
• J. Bourgain: Harmonic Analysis and Combinatorics: How Much May They Contribute to Each Other?, in Mathematics: Frontiers and Perspectives, V. Arnold, M. Atiyah, P. Lax, B. Mazur, eds., AMS 2000. An overwiev of the problems described above and the connections between them. Discusses recent work on Kakeya up to (and including) Bourgain's 1998 paper, but not the developments that followed.

• T. Wolff: Recent work connected with the Kakeya problem, in Prospects In Mathematics, H. Rossi, ed., AMS 1999. This article was written just a few months too early to include the additive number theory stuff. Nonetheless, if you are interested in the subject, you absolutely have to read it. Also includes a discussion of other problems in combinatorial geometry (Kakeya for circles, Falconer, Furstenberg, etc.).

• T. Tao: From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE, Notices of the AMS, March 2001, 294--303. Tao has since posted a statement on his web page, with a link to a second (more technical) statement. If you're like many of the Notices readers who have only a passing interest in Kakeya, you may well be wondering what the whole mess is about. I have therefore posted a few comments and corrections. Even the title is somewhat misleading. Connections between combinatorics etc. are not exactly "emerging" - they have been around for quite some time.

• E.M. Stein: Harmonic analysis, Princeton University Press, 1993. Although it obviously does not include the results obtained after 1993, it is still very much worth reading, just because its breadth of scope and perspective will not be beaten any time soon. Don't think of it as an introduction to the subject - get introduced to it somewhere else. But then do browse through Stein's book and you will discover plenty of gems that no-one else seems to remember.

• K. M. Davis and Y.-C. Chang, Lectures on Bochner-Riesz means, Cambridge University Press, 1987. I'm not sure if it is still in print. A very nice introduction to restriction, Bochner-Riesz, and some other related problems in harmonic analysis. Not as comprehensive as Stein's book, but it does not require much background and has the advantage of being shorter and easier to read.

• K.J. Falconer, The Geometry of Fractal Sets, Cambridge University Press 1985. The general background in geometric measure theory; includes a detailed discussion of Kakeya sets and some related problems. Very readable too.

• M. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer 1996. The background you might need if you decide to get interested in the additive number theory approach.

I am also preparing Tom Wolff's Caltech lecture notes for publication. A revised version should be available here in a few weeks, and the whole book (including bibliographical notes etc.) should be finished by the end of the summer. Last semester I gave a series of lectures at UBC based on Wolff's notes - many thanks to everyone who attended!

Several people, including Ben Green and Terence Tao, have posted expository Kakeya articles and lecture notes on their web pages. These are worth checking out, though I don't necessarily agree with everything they say.

If you need more details, you will have to look up the original research articles. Many of them are available online from the Los Alamos preprint archive at http://xxx.lanl.gov. They can also be downloaded from the UC Davis server at http://front.math.ucdavis.edu. Some time I might look up the links and put them here.

And did I say that there is plenty of ongoing research on these and related subjects? I will try to keep this page up-to-date and add new links and references as they appear. :-)

Last updated August 26, 2002.