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\begin{document}
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{\bf \LARGE  Recent progress on the Kakeya conjecture}

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Nets Katz (University of Illinois Chicago)\\

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Izabella {\L}aba (Princeton University)\\

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Terence Tao (UCLA)\\

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{\tt www.math.ucla.edu/$\sim$tao/preprints } (in preparation)
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\noindent
{\LARGE
The purpose of this talk is to
discuss recent progress on the Kakeya conjecture, including:
}
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\bi
 \item Bourgain's improvement in very high dimension ($n > 26$), and a further improvement to $n > 12$.

 \item A small improvement in the three-dimensional case

 \item A possible connection with the Falconer distance problem, Furstenburg problem, and Erd\"os's ring conjecture

\ei

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\noindent
What is the Kakeya conjecture?

\bi
\item Let $n \geq 2$.  A \emph{Besicovitch set} is defined as a subset of $\R^n$ which contains a unit line segment in every direction.  These sets can have measure zero (Besicovitch, 1918).

\item The \emph{Kakeya conjecture} states that all Besicovitch sets have dimension $n$.

\item One can phrase this for either the Minkowski or Hausdorff dimensions; this distinction will be irrelevant in all but one of the results.

\item  There is a more quantitative version of these conjectures involving the Kakeya maximal function
$$ f^*(\omega) = \sup_{l: l // \omega} \int_l f,$$
but we shall not discuss that here.

\item  These conjectures have application to various conjectures in harmonic analysis which involve ``wave packets'' that can point in arbitrary directions.  (Bochner-Riesz, restriction, and local smoothing conjectures, null form estimates, $\ldots$).

\ei

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Classical results

\bi

\item  In two dimensions the Kakeya conjecture is true (C\'ordoba, Davies, \ldots).  The idea is to count the number of pairs of intersecting lines, using the fact that any two lines intersect in at most one point.  (In higher dimensions this argument is not very efficient because most lines are skew).

\item  In higher dimension a Besicovitch set $E$ must have dimension at least $(n+1)/2$ (Christ, Drury, Duandikotxea, Rubio de Francia).  An informal version of this argument is as follows.  For each $\omega \in S^{n-1}$, let $l_\omega: [0,1] \to E$ be the unit line segment in $E$ which is parallel to $\omega$.  Define the map $\phi: S^{n-1} \times [0,1] \times [0,1] \to E \times E$ by
$$ \phi(\omega,s,t) = (l_\omega(s), l_\omega(t)).$$
This map is injective for $s \neq t$, since every two distinct points determine a unique line.  Since the domain of $\phi$ is $n+1$ dimensional, the range must have at least the same dimension, hence the claim.

\ei

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Improvements to the classical results.

\bi

\item  In 1991 Bourgain observed that one could improve upon the lower bound of $(n+1)/2$ in three and higher dimensions.  This was then followed by work of Wolff and Schlag; in 1995 Wolff showed the lower bound of $(n+2)/2$.  

\item Informally, the idea of Wolff's argument is as follows.  One first counts the average multiplicity of the unit line segments.  There are two cases,

\item If there is low multiplicity then the line segments do not overlap very much, and one can get a lower bound on the dimension of $E$.  

\item If there is high multiplicity, one considers the set of line segments which intersect a fixed line segment; such a set is known as a ``hairbrush''.  Because the multiplicity is high, this hairbrush has many bristles.  By dividing the hairbrush into two-dimensional planes around the stem line segment, and using C\'ordoba's argument on each plane, one obtains a lower bound on the dimension of the hairbrush, and thus a lower bound on $E$.  Balancing the two estimates give the result.

\ei

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Parallel lines

\bi

\item  One can try to generalize the notion of a Besicovitch set.  Call a \emph{generalized Besicovitch set} to be a subset of $\R^n$ which contains an $(n-1)$-dimensional family of unit line segments (some of which may be parallel).  To avoid degenerate cases (e.g. a unit square in $\R^3$), assume that there is at most a 1-dimensional sub-family contained in any 2-plane, etc.  One may still ask whether such sets must have dimension $n$.

\item  All the arguments mentioned earlier carry over to this setting.  It is not known whether this conjecture is true in higher dimensions, however there is an intriguing three-dimensional counterexample if the underlying field $\R$ is replaced by $\C$.  Namely, consider the Heisenberg group in $\C^3$:
$$ \{ (z_1, z_2, z_3): Im(z_3) = Im(z_1 \overline{z_2}) \}.$$
This set has real dimension 5 (so complex dimension 5/2), but contains a family of complex lines (i.e. planes) which has 4 real dimensions (so complex dimension 2).  This shows that Wolff's bound of $5/2$ in three dimension is sharp in the complex setting, or in any field which contains a sub-field of half the dimension.

\ei

\page

\bi

\item This raises an interesting question: do the reals $\R$ contain a subfield of dimension $1/2$?  The answer is negative (because the reals do not contain any non-trivial automorphisms).  But it is not known whether $\R$ contains
a 1/2 dimensional approximate field, that is a set $F$ of dimension $1/2$ such that the sets
$F + F$ and $F \cdot F$ are also $1/2$ dimensional (it can be shown that this implies that any rational combination of $F$ is also $1/2$ dimensional).  The existence of such a set would imply that Wolff's bound of $5/2$ is sharp for the \emph{generalized} Kakeya problem.

\item The existence of a 1/2 dimensional approximate field $F$ would also provide counterexamples to many other problems in combinatorial geometry.  For instance, the Falconer distance conjecture asserts that if a subset $E \subset \R^n$ has Hausdorff dimension 1 or more, then the distance set $\{ |x-y|: x,y \in E \}$ must also have dimension 1.  It is known that this is true if the set has dimension at least 4/3 (Wolff, 1998, following work by Falconer and Bourgain); but $E = F \times F$ would provide a counterexample to this conjecture.

\ei

\page

\bi

\item Another problem with a Kakeya flavor is the following problem of Furstenburg.  Let $E \subset \R^2$ be a set which contains a 1/2-dimensional subset of a line segment in every direction.  It's conjectured that $\dim(E) \geq 5/4$, but a variant of the $F \times F$ counterexample would have $\dim(E) = 1$.

\item All three problems mentioned above are equivalent (Katz, T.), in that improving the trivial bounds on one immediately leads to an improvement on the other two.

\item Related to this, there is an old and unsettled conjecture of Erd\"os which
asks whether $\R$ contains any 1/2 dimensional sub-rings, which would also be settled by any progress on the above problems.

\item Discrete versions of these problems are easier.  For instance, if $A$ is a finite subset of $R$, it's known that $\max(|A+A|,|A \cdot A|) \geq |A|^{5/4}$
(Elekes, 1995).

\ei
\page

Sums and differences

\bi
\item In 1998 Bourgain gave an argument based on arithmetic combinatorics which improves on Wolff's Kakeya bound $(n+2)/2$ for large $n$ ($n > 26$).  In fact he showed a bound of $13n/25 + 12/25$.

\item \textbf{Main Lemma.}  Let $A$, $B$, $C$ be finite subsets of an abelian group such that $|A|,|B|,|C| \leq N$, and let $G$ be a subset of $A \times B$ such that $a+b \in C$ for all $(a,b) \in G$, and the map $(a,b) \to a-b$ is one-to-one on $G$.  Then $|G| \leq N^{2-1/13}$.

\item $|G| \leq N^2$ is of course trivial.  The point is that if $G$ is too large, then $A+B$ is small while $A-B$ is large, which eventually leads to a contradiction.  

\item Assuming this lemma, the claim follows by considering three horizontal slices $A$, $C$, $B$ of a (discretized) Besicovitch set in arithmetic progression.  Every line segment in the set determines a pair in $A \times B$, and we let $G$ be the set of these pairs.  

\item Note this argument crucially uses the fact that the segments all point in different directions.  This is in contrast to previous work.
\ei

\page

\bi

\item The main lemma was originally proven by using some recent combinatorial machinery, in particular adapting some recent work by Gowers (1997) on the Balog-Szemeredi theorem.  The bound has been improved by an elementary argument to $N^{2-1/6}$ (Katz, T.),
which leads to a Kakeya bound of $6n/11 + 5/11$, which is new for $n > 12$.

\item The idea is as follows.  Define a vertical line segment $v$ to be a pair $v = ((a,b), (a',b))$ of elements of $G$ with the same $A$ co-ordinate, and define the length of this segment to be $|v| = a'-a$.  Similarly define the notion of horizontal line segments $h$.  We give upper and lower bounds on the quantity
$$ \{ (v,h): |v| = |h| \}.$$
The restriction $a+b\in C$ gives the lower bound, while the one-to-one nature of $a-b$ gives the upper bound.

\item On the other hand, there is a counterexample with $|G| = N^{log 6/log 3} = N^{2 - 0.36\ldots}$ (Rucza).

\ei

\page

The three-dimensional case

\bi 
\item  When $n=3$ the above arguments do not improve on Wolff's bound of $5/2$.  However, we have managed to improve this slightly to $5/2 + 10^{-10}$ for the
\emph{Minkowski} dimension of a Besicovitch set $E$ (Katz, Laba, T., 1999).

\item The idea is to assume that $E$ has dimension extremely close to $5/2$.  This implies a surprising amount of structure on $E$; eventually we have enough structure to obtain a contradiction.

\item The reason why we are restricted to the Minkowski dimension is as follows.  If $E$ has Minkowski dimension $5/2$, then we have $|N_\delta(E)| \lesssim \delta^{1/2}$ for \emph{all} $0 < \delta \ll 1$.  In the Hausdorff setting we have a similar bound for infinitely many $\delta$, but not all $\delta$.  (Also, we don't control all of $N_\delta(E)$, but only a sizeable fraction of it).
\ei

\page

First ingredient: stickiness

\bi 
\item Let $E$ be a Besicovitch set with Minkowski dimension exactly $5/2$.  In particular, we have $|N_\delta(E)| \lesssim \delta^{1/2}$ and $|N_{\sqrt{\delta}}(E)| \lesssim \delta^{1/4}$.

\item Since $E$ is a Besicovitch set, $N_\delta(E)$ contains $\delta^{-2}$ $\delta \times 1$ tubes in $\delta$-separated directions.  We say that this collection is \emph{sticky} if one can organize this collection into $\delta^{-1}$ bunches of $\delta^{-1}$ tubes each, such that each bunch is contained in a $\sqrt{\delta} \times 1$ tube.  (This is a kind of self-similarity condition).

\item One can show that most of the tubes have to have this stickiness property, because otherwise $N_{\sqrt{\delta}}(E)$ would be too large, thanks to an x-ray estimate of Wolff (1997).

\ei

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Fat tubes, thin tubes, short tubes

\bi
 \item Because of stickiness, the thin ($\delta \times 1$) tubes are contained inside some fat $(\sqrt{\delta} \times 1)$ tubes.  Inside each fat tube we have a bunch of thin tubes.  The behaviour of this bunch can be quite complicated (in fact, it's an affinely rescaled version of another Besicovitch set) but locally it's quite well behaved.  More precisely, the intersection of a bunch with a $\sqrt{\delta}$-ball looks like a union of parallel short $(\delta \times \sqrt{\delta})$ tubes.
\ei



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