LECTURE NOTES 8 FOR 254B

TERENCE TAO

1. An arithmetical approach to the Kakeya conjecture

In the past few lectures, we have been studying the Kakeya family of conjectures. For simplicity, let us now concentrate on the easiest conjecture in the whole family, namely the Minkowski dimension conjecture:

Conjecture 1.1. Let E be a Besicovitch set in R^{n}, i.e. a (compact) set which contains a unit line segment in every direction. Then the Minkowski dimension of E is equal to n.

By using various “geometric combinatorics” techniques, we have shown some progress on this conjecture, culminating in the 1995 result of Wolff, which showed that Besicovitch sets have to have Minkowski (and Hausdorff) dimension at least (n + 2)/2.

In 1998, Bourgain provided a completely different argument based on arithmetic combinatorics (the combinatorics of sums and differences, rather than points, lines, planes, etc.) which provides a rather different estimate. More precisely, he showed that Besicovitch sets have to have Hausdorff and Minkowski dimension at least 13n/25+12/25; this is superior to Wolff’s estimate for very high dimension n > 26. His arguments were based around some recent work of Tim Gowers; this work was (a very small part of) the reason why Gowers won the Fields medal last year.

We will show a more modest version of his result here, namely

Theorem 1.2. There exists an absolute constant c > 1/2 such that Besicovitch sets have Minkowski dimension at least cn + (1 − c).

This result is still superior to Wolff’s estimate if the dimension n is big enough.

We begin the proof. It is first convenient to make a technical change to the definition of a Besicovitch set. We parameterize R^{n} by x = (\underline{x}, x_{n}), where \underline{x} ∈ R^{n−1} and x_{n} ∈ R.

Definition 1.3. A modified Besicovitch set is a set E ⊂ R^{n−1} × [0, 1] which contains, for each w ∈ R^{n−1} such that |w| ≤ 1, a line segment of the form

\{(\underline{x}_{0} + wx_{n}, x_{n}) : x_{n} ∈ [0, 1]\}.

Exercise: Show that if there exists a Besicovitch set of Minkowski dimension d, then there exists a modified Besicovitch set of Minkowski dimension at most d.

Thus it suffices to show that modified Besicovitch sets have dimension at least cn + (1 − c), for some c > 1/2 to be chosen later.

Assume for contradiction that we can find a modified Besicovitch set E with Minkowski dimension at most cn + (1 − c). By the definition of Minkowski dimension, this implies that an upper bound on the neighbourhoods N_{δ}(E) for all 0 < δ ≪ 1, namely

|N_{δ}(E)| ≲ δ^{n−(cn+(1−c))} = δ^{(1−c)(n−1)}.

For each t ∈ [0, 1], let N_{δ}(E)[t] be the horizontal slice of N_{δ}(E) with the plane x_{n} = t. By Fubini’s theorem, we have

∫_{0}^{1} |N_{δ}(E)[t]| dt ≲ δ^{(1−c)(n−1)}.

In other words, |N_{δ}(E)[t]| is bounded by δ^{(1−c)(n−1)} on the average. By Chebyshev’s theorem, we therefore have

|\{t : |N_{δ}(E)[t]| > Cδ^{(1−c)(n−1)}\}| ≤ \frac{1}{100}

for sufficiently large C. In other words, if we let A denote the set

X = {t : |N_{δ}(E)[t]| ≤ Cδ^{(1−c)(n−1)},

then X occupies at least 99 percent of the unit interval.

As a consequence of this, X contains at least one arithmetic progression of length three s, s + d, s + 2d with spacing 1/10 < d. (To see this, pick any s in X ∩ [0, 1/10], and then let d range from 1/10 to 1/4, say. There are at most 10 percent of the d’s for which either s + d or s + 2d or both fail to be in X, hence the claim. Actually, X does not have to occupy such a huge portion of the interval to find decent-sized arithmetic progressions; a similar statement is true if X occupied only 1 percent of the interval (say), although it’s much more non-trivial to prove the claim then. The fact that arbitrarily low density sets contain arithmetic progressions of length three is due to Roth, and a beautiful ergodic theory proof was given by Furstenburg.)

By stretching the Besicovitch set some, and chopping off the bits below s or above s + 2d, we can assume that s = 0 and d = 1/2, so the progression is 0, 1/2, 1. (This doesn’t actually change very much, just makes the numbers nicer).

So we have three fairly small slices of our set N_{δ}(E), at x_{n} = 0, 1/2, 1. Now let’s discretize. Let C^{−1}δZ^{n−1} be the integer lattice of spacing C^{−1}δ for some large C, and let A[t] denote the sets

A[t] = {i ∈ C^{−1}δZ^{n−1} : (i, t) ∈ N_{δ}(E)}.

From the above claims and a simple covering argument, we thus see that

#A[0], #A[1/2], #A[1] ≲ δ^{−c(n−1)}

since each lattice point in A[t] corresponds to a n−1-dimensional δ-ball in N_{δ}(E)[t]

So we now have three fairly low-cardinality subsets of a lattice. Now let’s use the property that our set is a modified Besicovitch set. We know that E contains many line segments, so N_{δ}(E) contains many δ-tubes. Each such tube will intersect O(1) points in #A[0], #A[1/2], and #A[1]. Let G denote the subset of #A[0] × #A[1] which consists of pairs of such points which are contained in a single tube.

For any (x, y) ∈ G, the point (x + y)/2 is essentially in #A[1/2]. (Actually, it could be in one of O(1) translates of #A[1/2], but that’s not important). In particular, we have

#{x + y : (x, y) ∈ G} ≲ δ^{−c(n−1)}.

On the other hand, we know that the tubes all point in δ-separated directions, so that there are at least δ^{−(n−1)} tubes. This implies that

#{x − y : (x, y) ∈ G} ≳ δ^{−(n−1)}.

In other words, the sum set of G is small, yet the difference set of G is large. This is supposed to give a contradiction if c is too close to 1/2.

The desired result will follow if we can show the following lemma, which is a pure arithmetic combinatorics lemma.

Lemma 1.4. Let N ≫ 1, and let A and B be subsets of a free abelian group (e.g. C^{−1}δZ^{n}) with cardinality at most N. Let G be a subset of A × B such that

#{x + y : (x, y) ∈ G} ≤ N. (1)

Then we have

#{x − y : (x, y) ∈ G} ≲ N^{2−ε} (2)

for some absolute constant ε > 0.

Indeed, applying this lemma with N = δ^{−c(n−1)} we see that we are done if c < 1/(2 − ε). (Bourgain proved the above lemma with ε = 1/13, which explains the numerology c = 13/25).

Note that the claim is trivial if ε = 0, just by observing that

#{x − y : (x, y) ∈ G} ≤ #G ≤ #A#B ≤ N^{2};

the point is to somehow use the fact that the sum-set of G is small in order to get the difference set down as well.

2. Proof of main lemma

Okay, let’s begin the proof of the lemma. Let ε be extremely small, and assume for contradiction that we had a counterexample to the lemma for this value of ε. In other words, we found A, B, G such that

#{x − y : (x, y) ∈ G} ≈ N^{2} (3)

where we use A ⪅ B to denote the statement that A ≤ CN^{Cε}B. Observe that we may assume that

(4) (x, y) ↦ x − y is one to one on G,

since we can always refine G to make this the case. In particular, we have

#G ≈ N^{2}. (5)

In other words, G occupies a reasonable fraction of A × B.

Let’s view A × B as a lattice, with columns {a} × B and rows A × {b}. Let G^{b} denote the b^{th} row of G:

G^{b} = {a : (a, b) ∈ G}.

From (5) we know that

∑_{b} #G^{b} ≈ N^{2}. (6)

So on the average, most rows G^{b} contain about N elements. However, some rows may pull much less than their weight, so that #G^{b} ≪ N. But we shall simply eliminate such rows from B. Collectively, these rows contribute ≪ N^{2} to (6), so their removal will not affect (5) and thus (3). One can easily check that the other properties of the lemma are unaffected. We may therefore assume that

#G^{b} ≈ N for all b ∈ B. (7)

One could try to pull the same trick off for the columns and make them also size ≈ N uniformly, but there is a catch. Once we remove columns from A to enforce this, we lose the property (7) that we just gained. So it looks like we can only have one or the other. There are some ways to get both, but it’s not worth it for what we’re going to do.

We now try to exploit (1). Clearly, the sum-set of G is much smaller than G itself, which means that many pairs of elements of G, and hence in A × B have the same sum. From deep arithmetical fact:

a + b = a′ + b′ ⟺ a − b′ = a′ − b (8)

we thus expect many pairs of elements in A × B to have the same difference.

The quantitative version of this is as follows.

Lemma 2.1. Let A, B be sets of cardinality at most N, and let G ⊂ A × B be such that (1) and (5) holds. Then there exists a set I ⊂ A × B of cardinality

#I ≈ N^{2} (9)

such that for all (a, b) ∈ I we have

#{(a′, b′) ∈ A × B : a − b = a′ − b′} ≈ N.

In other words, there are many pairs (a, b) in A × B whose differences a − b are “popular”.

Proof Define the counting function c(x) by

c(x) = {(a, b) ∈ A × B : a + b = x}.

From (5) we have

∑_{x∈C} c(x) ≥ #G ≈ N^{2},

where C is the set in (1). On the other hand, from (1) and Cauchy-Schwarz we have

∑_{x∈C} c(x) ≤ #C^{1/2}(∑_{x} c(x)^{2})^{1/2} ≤ N^{1/2}(∑_{x} c(x)^{2})^{1/2}.

Thus we have

∑_{x} c(x)^{2} ⪆ N^{3},

which implies that

#{(a, b, a′, b′) ∈ A × B × A × B : a + b = a′ + b′} ⪆ N^{3}. (10)

By (8) we thus have

#{(a, b, a′, b′) ∈ A × B × A × B : a − b′ = a′ − b} ⪆ N^{3}.

If we now define the counting function by d(x) by

d(x) = {(a, b) ∈ A × B : a − b = x}

we get

∑_{x} d(x)^{2} ⪆ N^{3}.

On the other hand, since there are at most N^{2} pairs in A × B we have

∑_{x} d(x) ≤ N^{2}.

Thus we have

∑_{x:d(x)≪N} d(x)^{2} ⪅ N^{3},

and so

∑_{x:d(x)⪆N} d(x)^{2} ⪆ N^{3}.

On the other hand, we have d(x) ≤ N for every x, since for every a there is at most one b such that a − b = x. Thus we have

∑_{x:d(x)⪆N} N^{2} ⪆ N^{3}

so that

#{x : d(x) ⪆ N} ⪆ N.

If we let I be the set of all pairs (a, b) such that d(a − b) ⪆ N, we thus get the result.

Now let’s look at the situation. We have two subsets of A × B, namely G and I, with both occupy a large fraction of A × B, but have radically different properties. On the one hand, the differences a−b of elements of G are all distinct; on the other hand, the differences a−b of elements of I are very popular. Unfortunately, there is no reason why G and I should have any substantial intersection (unless the density of G is extremely high, such as 99%, but we want to handle more ambitious cases than this), so we seem to be at an impasse.

The trick that Bourgain used is as follows. As observed above, we can’t seem to engineer a contradiction by making the differences of G popular. However, it turns out, somewhat surprisingly, that it IS possible to make the differences of G popular in an iterated sense. More precisely, if (a, b) ∈ G, we cannot find many ways of writing a − b as a′ − b′ for some other pairs (a′, b′) ∈ A × B, but it turns out that we can very frequently write a − b as

a − b = (a_{1} − b_{1}) − (a_{2} − b_{2}) + (a_{3} − b_{3}) (11)

for a very large number (about N^{5}, in fact) of sextuplets (a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}) in A^{3} × B^{3}. Since there are only N^{6} such sextuplets, we thus see that there are only about N values of a−b, contradicting (3). At least, that’s the idea of the argument.

So, you ask, how could we possibly hope to show that a − b can be written in the form (11) in so many ways? The idea comes from another deep arithmetic identity:

For all a, a′, b, b′, a − b = (a − b′) − (a′ − b′) + (a′ − b).

Since a′ and b′ range over sets of cardinality about N, this gives a mere N^{2} representations of a − b. How do we bootstrap this to N^{5}? By arranging matters so that (a, b′), (a′, b′) and (a′, b) are usually in I, so that the differences a − b′, a′ − b′, and a′ − b are popular. Since popular differences can be represented about N different ways, we thus get a count of N^{5}.

There is still a large credibility gap in converting this argument into something rigorous. (There are a couple of notorious traps one can make in combinatorics that one has to avoid. For instance, if a property P occurs 50% of the time, and a property Q occurs 50% of the time, this does not imply that P and Q simultaneously hold 25%, or even 1% of the time. For instance, if we have N vertices, and half of all pairs of vertices are joined by an edge, this does not imply that there exists even a single triangle in this graph (think of the bipartite N/2, N/2 graph). One has to be careful when dealing with weasel words like “usually”, which I used a couple times above.) Let’s see how some of this is resolved.

As stated before, we want to arrange matters so that for many pairs of b, b′, there are many a′ for which (a′, b), (a′, b′) are both in I. We’ll do this as follows.

Definition 2.2. If b and b′ are elements of B, we say that b and b′ communicate, and write b ∼ b′, if we have

#{a′ ∈ A : (a′, b), (a′, b′) ∈ I} ≥ N^{1−ε_{0}}

where ε_{0} is an absolute constant to be chosen later.

We now show that we can make almost all the rows of A × B communicate with each other, if we first throw out some bad rows.

Lemma 2.3. There exists a subset B′ of B with cardinality #B′ ≈ N such that

#{b, b′ ∈ B′ : b ≁ b′} ⪅ N^{2−ε_{0}}. (12)

Proof The idea is to let B′ = I_{a} for some a ∈ A, where I_{a} is the column

I_{a} = {b : (a, b) ∈ I}.

The point of this is that

#{b, b′ ∈ I_{a} : b ≁ b′}

will be small on the average. Indeed, we see, by the magic of Fubini’s theorem, that

∑_{a∈A} #{b, b′ ∈ I_{a} : b ≁ b′} = ∑_{a∈A} ∑_{b,b′:b≁b′} χ_{I}(a, b)χ_{A}(a, b′)

= ∑_{b,b′:b≁b′} ∑_{a∈A} χ_{I}(a, b)χ_{A}(a, b′)

≤ ∑_{b,b′:b≁b′} N^{1−ε_{0}}

≤ N^{3−ε_{0}}.

On the other hand, from Cauchy-Schwarz we have

∑_{a∈A} #{b, b′ ∈ I_{a}} = ∑_{a∈A}(#I_{a})^{2}

≥ #A^{−1}(∑_{a∈A} #I_{a})^{2}

= (#I)^{2}/#A
⪆ N^{3}.

We therefore see that

∑_{a∈A} #{b, b′ ∈ I_{a}} − N^{ε_{0}−ε}#{b, b′ ∈ I_{a} : b ≁ b′} ⪆ N^{3}

for arbitrarily small ε (if N is sufficiently large). So there must exist an a such that

#{b, b′ ∈ I_{a}} ⪆ N^{ε_{0}}#{b, b′ ∈ I_{a} : b ≁ b′} + N^{2}.

It’s thus clear that B′ = I_{a} will do the trick for this lemma.

We now have a subset A×B′ of A×B with very communicative rows, and thus lots of elements of I. We have to make sure that this set still contains lots of elements of G, too, otherwise this exercise is in vain. Fortunately, we presciently ensured this way back when we obtained (7). We thus see that

#(G ∩ (A × B′)) = ∑_{b∈B′} #G^{b} ≈ N^{2}.

So we have no shortage of G points in A × B′.

Having trimmed the rows of A × B to get some good properties, we now turn to trimming the columns. The reason we want to do this is that we want to keep many elements of G on each column. If we define the (reduced) column G_{a} by

G_{a} = {b ∈ B′ : (a, b) ∈ G}

and define

A′ = {a ∈ A : #G_{a} ≈ N},

then a similar argument to the derivation of (7) shows that #A′ ≈ N and hence

#(G ∩ (A′ × B′)) ≈ N^{2},

so we can restrict the columns to the G-rich columns without losing too many elements of G.

Almost ready to finish the proof now. We apply Lemma 2.1 to G ∩ (A′ × B′), with A′ playing the role of A and B′ playing the role of B. This gives a set I′ ⊂ A′ × B′ of cardinality ≈ N^{2} such that for each (a, b) ∈ I′, the difference a−b can be written in ≈ N ways as a′ − b′ for some a′ ∈ A, b′ ∈ B.

Let us count the set

{(a, b, b′) ∈ A′ × B′ × B′ : (a, b) ∈ G, (a, b′) ∈ I′}.

There are ≈ N^{2} ways to choose (a, b′), and by the construction of A there are then ≈ N ways to choose b. So there are at least N^{3} triples here. On the other hand, consider the set

{(a, b, b′) ∈ A′ × B′ × B′ : (a, b) ∈ G, (a, b′) ∈ I′, b ≁ b′}.

By (12), there are ⪅ N^{2−ε_{0}} ways to choose b, b′, and then ≤ N ways to choose a. So there are ⪅ N^{3−ε_{0}} triples in this set. Therefore (if ε_{0} is small)

#{(a, b, b′) ∈ A′ × B′ × B′ : (a, b) ∈ G, (a, b′) ∈ I′, b ∼ b′} ⪆ N^{3}.

We rearrange this as

∑_{(a,b)∈G∩(A′×B′)} #{b′ : b ∼ b′, (a, b′) ∈ I′} ⪆ N^{3}.

There are at most N^{2} pairs (a, b) in the sum, and each sum is at most N. So there must exist ≈ N^{2} pairs (a, b) ∈ G ∩ (A′ × B′) such that

#{b′ : b ∼ b′, (a, b′) ∈ I′} ⪆ N.

By definition of communicativity, we therefore have

#{(a′, b′) : (a, b′) ∈ I′, (a′, b), (a′, b′) ∈ I} ⪆ N^{2−ε_{0}}

for these pairs (a, b). From the identity (11) and the popularity of I, I′, we therefore have

#{(a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}) : a − b = (a_{1} − b_{1}) − (a_{2} − b_{2}) + (a_{3} − b_{3})} ⪆ N^{5−ε_{0}}.

So we have ≈ N^{2} pairs (a, b) in G, whose differences a − b are each representable by (11) in ⪆ N^{5−ε_{0}} different ways. But there are only N^{6} ways to write something of the form (11). This contradicts (4) if ε_{0} is sufficiently small, and we are done.

As mentioned above, Bourgain went through this argument carefully and extracted a concrete value of ε = 1/13 from this argument. He extended the same result to Hausdorff dimension, by using a quantitative version of Roth’s theorem on arithmetic progressions of length three (there is one due to Heath-Brown and Szemeredi which suffices). The main difficulty with the Hausdorff dimension argument is that one does not get to consider sets such as N_{δ}(E) which fully contain many δ×1 tubes, but instead one gets various unions of δ-balls which only contain, say, 1/(log 1/δ)^{2} of a δ ×1 tube. This makes it harder to find arithmetic progressions, thus requiring the non-trivial combinatorial machinery.

What’s even more impressive is that Bourgain managed to push this further and get progress on the maximal operator conjecture (although now with a much weaker set of exponents). There the difficulty is even more pronounced; now one has to deal with sets which contain about λ of any given δ ×1 tube, where λ is quite small. Now one does not necessarily have three slices in arithmetic progression. However, it is still possible to find three slices whose reciprocals are in arithmetic progression, due to the curvature of the function x ↦ 1/x, and by invoking the wonders of projective geometry magic Bourgain was able to carry through the above type of argument to the maximal case.

This argument is still very new, and probably quite improvable. However the ε in Lemma 1.4 cannot be improved all the way to 1 (which would solve Kakeya). To see this, let n be a large number, and let A = B consist of those integers from 0 to 7^{n} − 1 whose base seven expansions only contain the digits 0, 1, and 3. Let G denote the pairs (a, b) of integers in A × B whose base seven expansions disagree at every one of the n digits. It is easy to see that A, B, and {a + b : (a, b) ∈ G} all have cardinality 3^{n}, yet {a − b : (a, b) ∈ G} has cardinality 6^{n}. This shows the upper bound ε ≤ 2 − log(6)/ log(3) = 0.36 . . .. I don’t know whether this is best possible.

3. Miscellaneous results

Due to lack of time, I wasn’t able to cover all the topics which I wanted to cover in this course. But I’ll talk briefly here about some other recent developments in the world of restriction and Kakeya problems. For a more complete survey of the latter topic I highly recommend the survey article of Wolff [42].

As we’ve seen a couple times already, it is sometimes beneficial to take a linear restriction estimate

∥f̂dσ∥_{p} ≲ ∥f∥_{q}

and replace it by a bilinear one such as

∥f̂dσĝdσ∥_{p/2} ≲ ∥f∥_{q}∥g∥_{q},

where we now impose some sort of separation condition on f and g (for instance, that their supports lie on sets whose separation is comparable to 1). Despite the more complicated appearance, these bilinear restriction estimates are easier to handle in many respects, and sometimes there are results which are only true in the bilinear setting^{1} For instance, we have the following very general estimate:

1On the other hand, in many cases the linear and bilinear estimates are essentially equivalent:

see [34].

Proposition 3.1. If dσ is surface measure on a smooth compact hypersurface S (possibly with boundary), and f and g live on disjoint subsets S_{1} and S_{2} of S, and are such that the normal directions of S_{1} and S_{2} are transverse to each other, then we have the bilinear estimate

∥f̂dσĝdσ∥_{2} ≲ ∥f∥_{2}∥g∥_{2} (13)

(cf. Lecture Notes 5).

Proof By Plancherel’s theorem it suffices to show that

∥fdσ ∗ gdσ∥_{2} ≲ ∥f∥_{2}∥g∥_{2}.

By Young’s inequality we have

∥fdσ ∗ gdσ∥_{1} ≲ ∥f∥_{1}∥g∥_{1}

so by interpolation it suffices to show that

∥fdσ ∗ gdσ∥_{∞} ≲ ∥f∥_{∞}∥g∥_{∞}.

Replacing everything by absolute values, we see that it suffices to show

∥χ_{S_{1}}dσ ∗ χ_{S_{2}}dσ∥_{∞} ≲ 1,

but this follows from the transversality assumption on S_{1} and S_{2}. (One can make this rigorous in many ways, for instance approximating the measure χ_{S_{1}}dσ as the weak limit of functions concentrated near S_{1}.)

In particular, (13) holds when n = 2 for the sphere S^{1}. However, the corresponding linear estimate

∥f̂dσ∥_{4} ≲ ∥f∥_{2},

which would have easily implied the above bilinear estimate, is false (using the Knapp example, see Lecture Notes 1). The best exponent one can place on the left-hand side here is just 6.

More generally, the Tomas-Stein theorem asserts that in n dimensions and for the sphere S^{n−1}, we have the estimate

∥f̂dσ∥_{p} ≲ ∥f∥_{2}

whenever p ≥ 2(n + 1)/(n − 1), and this is best possible. However, if we go to the bilinear setting, it is conjectured that the corresponding estimate

∥f̂dσĝdσ∥_{p/2} ≲ ∥f∥_{2}∥g∥_{2} (14)

holds for a larger range of p, assuming of course the separation condition on f and g; in fact Machedon and Klainerman have conjectured that this estimate holds for the larger range p ≥ 2(n + 2)/n.

This range is sharp. To see this, consider two δ × δ^{1/2} × … δ^{1/2} rectangles C_{1}, C_{2} on the sphere (“squashed caps”, since this is a usual δ^{1/2} cap with one dimension squashed down to δ), oriented so that their n − 2 long directions are parallel. Let f and g be characteristic functions of these caps. Then one can verify (Exercise!) that |f̂dσ(x)| ∼ |ĝdσ(x)| ∼ δ^{n/2} on an δ^{−1} × δ^{−1} × δ^{−1/2} × … × δ^{−1/2} box in R^{n} centered at the origin. This is basically because C_{1} and C_{2} can both be placed inside parallel δ × δ × δ^{1/2} × … × δ^{1/2} boxes. Doing the arithmetic we get the bound on p.

As we saw above, the Machedon-Klainerman conjecture is true in two dimensions. In three dimensions the best result is that (14) is true for all p > 2 − \frac{2}{17} [35]; this is a slight improvement on what one can get from Tomas-Stein, which is p ≥ 2. For comparison, the conjecture in three dimensions is p ≥ 2 − \frac{1}{3}. The estimate is obtained by taking local and global (bilinear) restriction theorems, as in the previous set of notes, and interpolating it with the following improvement of (13) in three dimensions for the sphere S^{2}:

∥f̂dσĝdσ∥_{2} ≲ ∥f∥_{12/7}∥g∥_{12/7}. (15)

This estimate is due to Moyua, Vargas, and Vega [22], and is proven by expanding the left-hand side using Plancherel’s theorem, which gives a convolution of two spherical measures as before, but then refusing to simply obtain an estimate by interpolation between L^{1} and L^{∞} estimates. Instead, they apply the machinery of Radon transform-like operators, which would take us too far afield for these notes. Incidentally, (15) can be obtained by interpolating the endpoint Machedon-Klainerman conjecture with the trivial estimate

∥f̂dσĝdσ∥_{∞} ≲ ∥f∥_{1}∥g∥_{1}.

An extremely recent development (March 1999) is that the Machedon-Klainerman conjecture has been settled for the cone in all dimensions, i.e. instead of considering the sphere S^{n−1} in R^{n}, we consider the cone

{(ξ, |ξ|) : ξ ∈ R^{n}, |ξ| ∼ 1}

in R^{n+1}. (Despite the extra dimension, the numerology is the same; informally, the surface has picked up a flat direction, which is not expected to contribute to the exponents. There is no good formal relationship between restriction theorems for the cone and those of the sphere, however.) This is especially exciting, given that the Fourier transforms of measures on the cone are precisely what needs to be studied in the theory of linear and non-linear wave equations. (The sphere is of course also interesting for PDE, as it corresponds to the Helmholtz equation, and the closely related paraboloid is related to the Schrödinger equation as we’ve already noted in Lecture notes 3). Currently, the estimates used to control nonlinear wave equations are based on L^{2} estimates which are basically variants of Proposition 3.1.

More precisely, Wolff [43] has shown (14) for the cone in the near-optimal range p > 2(n+2)/n. This has recently been sharpened [37] to p ≥ 2(n+2)/n by refining Wolff’s argument.

The following sketch will do terrible injustice to Wolff’s argument. By a variant [3], [35] of the machinery developed in the previous set of Notes, it suffices to prove the local bilinear restriction estimate

∥f̂dσĝdσ∥_{L^{p/2}(B(0,R))} ≲ R^{ε}∥f∥_{2}∥g∥_{2}

for all large R and small ε, with the constant dependent on ε.

We break up fdσ into pieces supported on 1/\sqrt{R} sectors, which are the analogues of 1/\sqrt{R} caps as used in the previous set of notes, and churn away as before. We soon find ourselves having to deal with expressions like

(∑_{T} χ_{T})(∑_{T′} χ_{T′})

where the T are \sqrt{R} × R tubes pointing in a set of null directions, and T′ are similarly \sqrt{R} × R tubes which point in a different set of null directions (so that the angle between T and T′ is always about 1 - cf. the quantities studied in Lecture Notes 6). (In general, the direction the tube points in is the normal of the surface at the location in frequency space that the tube corresponds to. For the sphere there are normals in all directions, and thus tubes in all directions, but for the cone the only normals are those in null directions.)

We can organize the tubes T around their points of high multiplicity, thus creating various “bushes” - many T -tubes passing through a single point. Now the other set of tubes, the T′, can intersect this bush in one of two ways: either it can hit the bush near the center of the bush, or it can hit it on the periphery.

If it happens that the T′ tubes usually hit the T bushes near the center of these bushes, then it turns out that it is beneficial to decompose the R-ball B(0, R) into smaller balls, e.g. R^{1−ε} balls, which cover the region near the centers of the bushes and which thus capture most of the action. An induction argument then takes care of this case.

Now suppose that the tubes T′ usually hit the T -bushes on the periphery. (There is a third case, when the tubes T′ hardly hit the T -bushes at all, but this case is easy). In this case the key observation is the following: if a tube T′ hits a T -bush on the periphery, then it only hits O(1) of the T -tubes in this bush. This is a special property of null directions - basically, it says that no three null directions ever lie in a plane - and it is completely and utterly false in the spherical case. (So Wolff’s result does not appear to be replicable in the spherical case, or at least the argument is insufficient to resolve the entire conjecture). This observation limits the number of incidences between T and T′ by exactly the amount needed to get a sharp result.

There is one final ingredient in Wolff’s argument, just as in the previous Lecture notes. The Kakeya arguments limit the number of \sqrt{R}-cubes which come into play, and limit how many tubes interact with each \sqrt{R} cube, but we still need to estimate the L^{p} norm of f̂dσĝdσ on these cubes. This is done by interpolating between an easy L^{1} estimate (basically, Cauchy-Schwarz combined with the estimate R^{∗}(2 → 2, 1/2)), together with an L^{2} estimate which is a more sophisticated version of Proposition 3.1. (Quantities like ∥f̂dσĝdσ∥_{2}, or ∥f̂dσ∥_{4} have a long history, both in the world of harmonic analysis (see e.g. [18], [19], [20], [3] [36]) and in PDE (see e.g. [13], [12], [14]), and have been extensively studied).

As one corollary to Wolff’s argument, one can also prove linear restriction estimates for the cone. In particular, the restriction conjecture for cones (which states that ∥fdσ∥_{p} ≲ ∥f∥_{q} holds for the unit cone in R^{n+1} whenever p > 2n/(n − 1) and p ≥ (n + 1)q′/(n − 1)) has been settled by Wolff in the four-dimensional case. (The two dimensional case is a triviality, and the three dimensional case was done by Barcelo [1]).

Another area of recent research has been in extending the Kakeya estimates proven earlier. In the negative direction, Chris Sogge and Bill Minicozzi have shown that curved space analogues of (for instance) Wolff’s maximal estimates can fail, even in three dimensions. The difficulty is that there is no analogue of statements such as “any two intersecting lines are contained in exactly one plane” in curved space, and indeed it is possible for many geodesics to converge upon a single surface.

On the other hand, if one replaces lines by circles, and rectangles by annuli, Wolff and Kolasa [40], [15] have managed to get sharp estimates for the corresponding maximal functions. (See also some later work by Schlag [23], [24]). One interesting feature of their argument is the incorporation of some sophisticated combinatorial tools (i.e. more than just Fubini, Cauchy-Schwarz, and the pigeonhole principle). In particular, the arguments employ the method of cells, which was originally developed for discrete problems such as the incidence problem (given n points and m lines in the plane, what is the maximal number of point-line incidences?) or Erdös’s distance problem (given n points in the plane, what is the smallest number of distinct distances between these points?). Incidentally, the latter problem is still unsolved (the conjecture is at least ≳ n^{1−ε} distances for all ε, although the best record to date is ≳ n^{4/5−ε}), and I consider it quite interesting.

Wolff [41] has also extended his Kakeya maximal operator estimate in an important direction. Recall that this estimate was of the form

∥ ∑_{T} χ_{T} ∥_{p} ⪅ δ^{\frac{n}{p} −(n−1)},

where the tubes T were δ × 1 tubes pointing in a δ-separated set of directions. The question then arises of what one can say if one does not assume that the tubes all point in different directions; for instance, suppose that one allows as many as m tubes pointing in the same direction for some m ≥ 1. (We consider two tubes to point in the same direction if they are within δ of each other in orientation). To avoid degenerate cases, we assume that the m parallel tubes in each direction are still δ-separated in position, if not in angle. From the triangle inequality one can get the trivial estimate

∥ ∑_{T} χ_{T} ∥_{p} ⪅ mδ^{\frac{n}{p} −(n−1)},

but one expects to improve upon this. For instance, when p = ∞ we have no dependence of m at all:

∥ ∑_{T} χ_{T} ∥_{∞} ≲ δ^{−(n−1)};

this just reflects the fact that every point is contained in O(δ^{−(n−1)}) tubes, regardless of whether one assumes the tubes are pointing in different directions or not. On the other extreme, at the conjectured exponent p = n/(n − 1) one does not expect any improvement over the triangle inequality. For intermediate values of p, interpolation thus suggests the bound

∥ ∑_{T} χ_{T} ∥_{p} ⪅ m^{n/p}δ^{\frac{n}{p} −(n−1)}. (16)

The good news is that this bound has been proven in [41] - for three dimensions, at least - for the same range of exponents as Wolff’s Kakeya estimate, namely p ≥ (n + 2)/n. (Recently, Izabella Laba and I [16] have partially extended this result to higher dimensions). These types of bounds are known as x-ray transform estimates, as they can be rephrased as smoothing estimates on the x-ray transform

Xf(l) = ∫_{l} f

which takes functions on R^{n} to functions on G(n, 1), the space of lines in R^{n}. This improved estimate comes in handy in a number of places, for instance it is a necessary ingredient in the current world record for restriction conjectures for the sphere [35].

The argument is a variant of the hairbrush argument we discussed in Lecture Notes 6, but with a twist. Basically, if we have many tubes in the same direction we begin to encounter objects called “plates”, which are a large number of δ × 1 tubes compressed into what is essentially a δ×ω×1 box for some δ ≤ ω ≤ 1. The presence of these plates causes the Córdoba argument - which is a key part of Wolff’s original Kakeya argument - to give poor estimates. However, if there are large plates lying around, we can use them as the stem of a hairbrush, thus obtaining a thicker (and thus larger in measure) set than the one in Wolff’s original argument. Playing off the gain against the loss, we end up with (16).

As remarked at the beginning of these notes, the recent arguments of Bourgain have made progress to the Kakeya conjecture in high dimensions, but for low dimensions - in particular in three dimensions - Wolff’s argument is still superior, giving the lower bound of 5/2 for the Minkowski dimension of Besicovitch sets in R^{3}. Recently Katz, Laba, and myself [11] have managed to combine the two arguments with some other lines of reasoning, and improve this bound marginally to 5/2 + 10^{−10}. For several reasons, this argument is restricted to the Minkowski dimension, and there are serious difficulties in getting to the Hausdorff dimension, let alone the maximal function problem, with this argument.

Very rapidly, the argument (which is unfortunately quite involved) runs as follows. Suppose for contradiction that we had a Besicovitch set E in R^{3} whose Minkowski dimension was essentially 5/2. Then the δ-neighbourhood of E is a collection of δ×1 tubes compressed into a set of size about δ^{1/2}, while the \sqrt{δ}-neighbourhood of E is a collection of \sqrt{δ}×1 tubes compressed into a set of size about δ^{1/4}. Putting the two facts together - and crucially using Wolff’s x-ray estimate (16) - we see that there is a nesting relationship between the δ tubes and the \sqrt{δ} tubes. Namely, every \sqrt{δ}-tube contains about δ^{−1} δ-tubes, which is the maximal number of directionseparated tubes it is allowed to contain. This property was first observed by Wolff [41], and is called “stickiness”.

Now we fix a single \sqrt{δ} tube, and look at the δ tubes lying inside it. If we divide this \sqrt{δ} tube into \sqrt{δ}-cubes, we see on each cube that the collection of δ tubes actually just looks like a collection of parallel δ ×δ ×\sqrt{δ} tubes, oriented in the same direction as the large \sqrt{δ} tube.

Now we consider the situation of three \sqrt{δ}-tubes intersecting in a common \sqrt{δ}- cube; this situation has to occur quite often. We thus get three separate families of parallel δ × δ × \sqrt{δ} tubes, each pointing in a different direction. Either these families overlap highly or they don’t. If the overlap is small, then their union will be large, and it is possible to contradict the assumption that the δ-neighbourhood of E is compressed into a set of size δ^{1/2}. Thus the overlap must be high. But it turns out that this can only happen if the three directions of these families of δ × δ × \sqrt{δ} tubes are essentially coplanar.

Unraveling what this means, this implies that all the \sqrt{δ}-tubes through a point actually lie in a plane. This property we call “planiness”. By a rescaling argument one can also ensure that all the δ-tubes through a point lie in a plane.

The planiness property, together with a closely allied property which we call “graininess”, severely restrict the degrees of freedom of the Besicovitch set. By coupling this additional structural information with Bourgain’s argument, we were able to improve Bourgain’s dimensional estimate from its current value of 2\frac{1}{25} to \frac{5}{2} + ε for some small ε > 0, which concludes the proof.

There are many other facets of these questions which are still being studied, with many interesting developments and insights. One area which I have devoted very little time to is the study of Bochner-Riesz multipliers. These multipliers are closely related to restriction problems (for a formal relationship between these problems, see [32]), but contain some additional difficulties, especially when one considers questions of pointwise convergence (since one then has to control a maximal oscillatory operator rather than just a single linear operator). I won’t say too much about developments here, but for some representative work on this problem I cite the results of Carbery [6], [7], Christ [8], [10], [9], and myself [30], [31], [33]. Other authors, notably Seeger [26], [27] and Sogge [28], [29], have also studied the curved space case, which has its own special difficulties.

This is still only a very partial list of the work and effort which has been put into these problems. For instance, much work as been put into Kakeya problems with a restricted set of directions by such people as Córdoba, Katz, and Barrionuevo. The amount of effort poured into these problems should give some idea of the difficulty, richness and interest of these deceptively simple questions.

Have a good summer!

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Department of Mathematics, UCLA, Los Angeles CA 90095-1555

E-mail address: tao@math.ucla.edu