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This is a blog version of a talk I recently gave at the IPAM workshop on “The Kakeya Problem, Restriction Problem, and Sum-product Theory”.

Note: the discussion here will be highly non-rigorous in nature, being extremely loose in particular with asymptotic notation and with the notion of dimension. Caveat emptor.

One of the most infamous unsolved problems at the intersection of geometric measure theory, incidence combinatorics, and real-variable harmonic analysis is the Kakeya set conjecture. We will focus on the following three-dimensional case of the conjecture, stated informally as follows:

Conjecture 1 (Kakeya conjecture) Let ${E}$ be a subset of ${{\bf R}^3}$ that contains a unit line segment in every direction. Then ${\hbox{dim}(E) = 3}$.

This conjecture is not precisely formulated here, because we have not specified exactly what type of set ${E}$ is (e.g. measurable, Borel, compact, etc.) and what notion of dimension we are using. We will deliberately ignore these technical details in this post. It is slightly more convenient for us here to work with lines instead of unit line segments, so we work with the following slight variant of the conjecture (which is essentially equivalent):

Conjecture 2 (Kakeya conjecture, again) Let ${{\cal L}}$ be a family of lines in ${{\bf R}^3}$ that meet ${B(0,1)}$ and contain a line in each direction. Let ${E}$ be the union of the restriction ${\ell \cap B(0,2)}$ to ${B(0,2)}$ of every line ${\ell}$ in ${{\cal L}}$. Then ${\hbox{dim}(E) = 3}$.

As the space of all directions in ${{\bf R}^3}$ is two-dimensional, we thus see that ${{\cal L}}$ is an (at least) two-dimensional subset of the four-dimensional space of lines in ${{\bf R}^3}$ (actually, it lies in a compact subset of this space, since we have constrained the lines to meet ${B(0,1)}$). One could then ask if this is the only property of ${{\cal L}}$ that is needed to establish the Kakeya conjecture, that is to say if any subset of ${B(0,2)}$ which contains a two-dimensional family of lines (restricted to ${B(0,2)}$, and meeting ${B(0,1)}$) is necessarily three-dimensional. Here we have an easy counterexample, namely a plane in ${B(0,2)}$ (passing through the origin), which contains a two-dimensional collection of lines. However, we can exclude this case by adding an additional axiom, leading to what one might call a “strong” Kakeya conjecture:

Conjecture 3 (Strong Kakeya conjecture) Let ${{\cal L}}$ be a two-dimensional family of lines in ${{\bf R}^3}$ that meet ${B(0,1)}$, and assume the Wolff axiom that no (affine) plane contains more than a one-dimensional family of lines in ${{\cal L}}$. Let ${E}$ be the union of the restriction ${\ell \cap B(0,2)}$ of every line ${\ell}$ in ${{\cal L}}$. Then ${\hbox{dim}(E) = 3}$.

Actually, to make things work out we need a more quantitative version of the Wolff axiom in which we constrain the metric entropy (and not just dimension) of lines that lie close to a plane, rather than exactly on the plane. However, for the informal discussion here we will ignore these technical details. Families of lines that lie in different directions will obey the Wolff axiom, but the converse is not true in general.

In 1995, Wolff established the important lower bound ${\hbox{dim}(E) \geq 5/2}$ (for various notions of dimension, e.g. Hausdorff dimension) for sets ${E}$ in Conjecture 3 (and hence also for the other forms of the Kakeya problem). However, there is a key obstruction to going beyond the ${5/2}$ barrier, coming from the possible existence of half-dimensional (approximate) subfields of the reals ${{\bf R}}$. To explain this problem, it easiest to first discuss the complex version of the strong Kakeya conjecture, in which all relevant (real) dimensions are doubled:

Conjecture 4 (Strong Kakeya conjecture over ${{\bf C}}$) Let ${{\cal L}}$ be a four (real) dimensional family of complex lines in ${{\bf C}^3}$ that meet the unit ball ${B(0,1)}$ in ${{\bf C}^3}$, and assume the Wolff axiom that no four (real) dimensional (affine) subspace contains more than a two (real) dimensional family of complex lines in ${{\cal L}}$. Let ${E}$ be the union of the restriction ${\ell \cap B(0,2)}$ of every complex line ${\ell}$ in ${{\cal L}}$. Then ${E}$ has real dimension ${6}$.

The argument of Wolff can be adapted to the complex case to show that all sets ${E}$ occuring in Conjecture 4 have real dimension at least ${5}$. Unfortunately, this is sharp, due to the following fundamental counterexample:

Proposition 5 (Heisenberg group counterexample) Let ${H \subset {\bf C}^3}$ be the Heisenberg group

$\displaystyle H = \{ (z_1,z_2,z_3) \in {\bf C}^3: \hbox{Im}(z_1) = \hbox{Im}(z_2 \overline{z_3}) \}$

and let ${{\cal L}}$ be the family of complex lines

$\displaystyle \ell_{s,t,\alpha} := \{ (\overline{\alpha} z + t, z, sz + \alpha): z \in {\bf C} \}$

with ${s,t \in {\bf R}}$ and ${\alpha \in {\bf C}}$. Then ${H}$ is a five (real) dimensional subset of ${{\bf C}^3}$ that contains every line in the four (real) dimensional set ${{\cal L}}$; however each four real dimensional (affine) subspace contains at most a two (real) dimensional set of lines in ${{\cal L}}$. In particular, the strong Kakeya conjecture over the complex numbers is false.

This proposition is proven by a routine computation, which we omit here. The group structure on ${H}$ is given by the group law

$\displaystyle (z_1,z_2,z_3) \cdot (w_1,w_2,w_3) = (z_1 + w_1 + z_2 \overline{w_3} - z_3 \overline{w_2}, z_2 +w_2, z_3+w_3),$

giving ${E}$ the structure of a ${2}$-step simply-connected nilpotent Lie group, isomorphic to the usual Heisenberg group over ${{\bf R}^2}$. Note that while the Heisenberg group is a counterexample to the complex strong Kakeya conjecture, it is not a counterexample to the complex form of the original Kakeya conjecture, because the complex lines ${{\cal L}}$ in the Heisenberg counterexample do not point in distinct directions, but instead only point in a three (real) dimensional subset of the four (real) dimensional space of available directions for complex lines. For instance, one has the one real-dimensional family of parallel lines

$\displaystyle \ell_{0,t,0} = \{ (t, z, 0): z \in {\bf C}\}$

with ${t \in {\bf R}}$; multiplying this family of lines on the right by a group element in ${H}$ gives other families of parallel lines, which in fact sweep out all of ${{\cal L}}$.

The Heisenberg counterexample ultimately arises from the “half-dimensional” (and hence degree two) subfield ${{\bf R}}$ of ${{\bf C}}$, which induces an involution ${z \mapsto \overline{z}}$ which can then be used to define the Heisenberg group ${H}$ through the formula

$\displaystyle H = \{ (z_1,z_2,z_3) \in {\bf C}^3: z_1 - \overline{z_1} = z_2 \overline{z_3} - z_3 \overline{z_2} \}.$

Analogous Heisenberg counterexamples can also be constructed if one works over finite fields ${{\bf F}_{q^2}}$ that contain a “half-dimensional” subfield ${{\bf F}_q}$; we leave the details to the interested reader. Morally speaking, if ${{\bf R}}$ in turn contained a subfield of dimension ${1/2}$ (or even a subring or “approximate subring”), then one ought to be able to use this field to generate a counterexample to the strong Kakeya conjecture over the reals. Fortunately, such subfields do not exist; this was a conjecture of Erdos and Volkmann that was proven by Edgar and Miller, and more quantitatively by Bourgain (answering a question of Nets Katz and myself). However, this fact is not entirely trivial to prove, being a key example of the sum-product phenomenon.

We thus see that to go beyond the ${5/2}$ dimension bound of Wolff for the 3D Kakeya problem over the reals, one must do at least one of two things:

• (a) Exploit the distinct directions of the lines in ${{\mathcal L}}$ in a way that goes beyond the Wolff axiom; or
• (b) Exploit the fact that ${{\bf R}}$ does not contain half-dimensional subfields (or more generally, intermediate-dimensional approximate subrings).

(The situation is more complicated in higher dimensions, as there are more obstructions than the Heisenberg group; for instance, in four dimensions quadric surfaces are an important obstruction, as discussed in this paper of mine.)

Various partial or complete results on the Kakeya problem over various fields have been obtained through route (a) or route (b). For instance, in 2000, Nets Katz, Izabella Laba and myself used route (a) to improve Wolff’s lower bound of ${5/2}$ for Kakeya sets very slightly to ${5/2+10^{-10}}$ (for a weak notion of dimension, namely upper Minkowski dimension). In 2004, Bourgain, Katz, and myself established a sum-product estimate which (among other things) ruled out approximate intermediate-dimensional subrings of ${{\bf F}_p}$, and then pursued route (b) to obtain a corresponding improvement ${5/2+\epsilon}$ to the Kakeya conjecture over finite fields of prime order. The analogous (discretised) sum-product estimate over the reals was established by Bourgain in 2003, which in principle would allow one to extend the result of Katz, Laba and myself to the strong Kakeya setting, but this has not been carried out in the literature. Finally, in 2009, Dvir used route (a) and introduced the polynomial method (as discussed previously here) to completely settle the Kakeya conjecture in finite fields.

Below the fold, I present a heuristic argument of Nets Katz and myself, which in principle would use route (b) to establish the full (strong) Kakeya conjecture. In broad terms, the strategy is as follows:

1. Assume that the (strong) Kakeya conjecture fails, so that there are sets ${E}$ of the form in Conjecture 3 of dimension ${3-\sigma}$ for some ${\sigma>0}$. Assume that ${E}$ is “optimal”, in the sense that ${\sigma}$ is as large as possible.
2. Use the optimality of ${E}$ (and suitable non-isotropic rescalings) to establish strong forms of standard structural properties expected of such sets ${E}$, namely “stickiness”, “planiness”, “local graininess” and “global graininess” (we will roughly describe these properties below). Heuristically, these properties are constraining ${E}$ to “behave like” a putative Heisenberg group counterexample.
3. By playing all these structural properties off of each other, show that ${E}$ can be parameterised locally by a one-dimensional set which generates a counterexample to Bourgain’s sum-product theorem. This contradiction establishes the Kakeya conjecture.

Nets and I have had an informal version of argument for many years, but were never able to make a satisfactory theorem (or even a partial Kakeya result) out of it, because we could not rigorously establish anywhere near enough of the necessary structural properties (stickiness, planiness, etc.) on the optimal set ${E}$ for a large number of reasons (one of which being that we did not have a good notion of dimension that did everything that we wished to demand of it). However, there is beginning to be movement in these directions (e.g. in this recent result of Guth using the polynomial method obtaining a weak version of local graininess on certain Kakeya sets). In view of this (and given that neither Nets or I have been actively working in this direction for some time now, due to many other projects), we’ve decided to distribute these ideas more widely than before, and in particular on this blog.

Combinatorial incidence geometry is the study of the possible combinatorial configurations between geometric objects such as lines and circles. One of the basic open problems in the subject has been the Erdős distance problem, posed in 1946:

Problem 1 (Erdős distance problem) Let ${N}$ be a large natural number. What is the least number ${\# \{ |x_i-x_j|: 1 \leq i < j \leq N \}}$ of distances that are determined by ${N}$ points ${x_1,\ldots,x_N}$ in the plane?

Erdős called this least number ${g(N)}$. For instance, one can check that ${g(3)=1}$ and ${g(4)=2}$, although the precise computation of ${g}$ rapidly becomes more difficult after this. By considering ${N}$ points in arithmetic progression, we see that ${g(N) \leq N-1}$. By considering the slightly more sophisticated example of a ${\sqrt{N} \times \sqrt{N}}$ lattice grid (assuming that ${N}$ is a square number for simplicity), and using some analytic number theory, one can obtain the slightly better asymptotic bound ${g(N) = O( N / \sqrt{\log N} )}$.

On the other hand, lower bounds are more difficult to obtain. As observed by Erdős, an easy argument, ultimately based on the incidence geometry fact that any two circles intersect in at most two points, gives the lower bound ${g(N) \gg N^{1/2}}$. The exponent ${1/2}$ has been slowly increasing over the years by a series of increasingly intricate arguments combining incidence geometry facts with other known results in combinatorial incidence geometry (most notably the Szemerédi-Trotter theorem) and also some tools from additive combinatorics; however, these methods seemed to fall quite short of getting to the optimal exponent of ${1}$. (Indeed, previously to last week, the best lower bound known was approximately ${N^{0.8641}}$, due to Katz and Tardos.)

Very recently, though, Guth and Katz have obtained a near-optimal result:

Theorem 2 One has ${g(N) \gg N / \log N}$.

The proof neatly combines together several powerful and modern tools in a new way: a recent geometric reformulation of the problem due to Elekes and Sharir; the polynomial method as used recently by Dvir, Guth, and Guth-Katz on related incidence geometry problems (and discussed previously on this blog); and the somewhat older method of cell decomposition (also discussed on this blog). A key new insight is that the polynomial method (and more specifically, the polynomial Ham Sandwich theorem, also discussed previously on this blog) can be used to efficiently create cells.

In this post, I thought I would sketch some of the key ideas used in the proof, though I will not give the full argument here (the paper itself is largely self-contained, well motivated, and of only moderate length). In particular I will not go through all the various cases of configuration types that one has to deal with in the full argument, but only some illustrative special cases.

To simplify the exposition, I will repeatedly rely on “pigeonholing cheats”. A typical such cheat: if I have ${n}$ objects (e.g. ${n}$ points or ${n}$ lines), each of which could be of one of two types, I will assume that either all ${n}$ of the objects are of the first type, or all ${n}$ of the objects are of the second type. (In truth, I can only assume that at least ${n/2}$ of the objects are of the first type, or at least ${n/2}$ of the objects are of the second type; but in practice, having ${n/2}$ instead of ${n}$ only ends up costing an unimportant multiplicative constant in the type of estimates used here.) A related such cheat: if one has ${n}$ objects ${A_1,\ldots,A_n}$ (again, think of ${n}$ points or ${n}$ circles), and to each object ${A_i}$ one can associate some natural number ${k_i}$ (e.g. some sort of “multiplicity” for ${A_i}$) that is of “polynomial size” (of size ${O(N^{O(1)})}$), then I will assume in fact that all the ${k_i}$ are in a fixed dyadic range ${[k,2k]}$ for some ${k}$. (In practice, the dyadic pigeonhole principle can only achieve this after throwing away all but about ${n/\log N}$ of the original ${n}$ objects; it is this type of logarithmic loss that eventually leads to the logarithmic factor in the main theorem.) Using the notation ${X \sim Y}$ to denote the assertion that ${C^{-1} Y \leq X \leq CY}$ for an absolute constant ${C}$, we thus have ${k_i \sim k}$ for all ${i}$, thus ${k_i}$ is morally constant.

I will also use asymptotic notation rather loosely, to avoid cluttering the exposition with a certain amount of routine but tedious bookkeeping of constants. In particular, I will use the informal notation ${X \lll Y}$ or ${Y \ggg X}$ to denote the statement that ${X}$ is “much less than” ${Y}$ or ${Y}$ is “much larger than” ${X}$, by some large constant factor.