AN INFORMAL PROOF OF WOLFF’S X-RAY ESTIMATE

TERENCE TAO

Abstract. We give an informal proof of Wolff’s x-ray estimate [5].

1. Introduction

Let 0 < δ ≪ 1 be a small number and 1 ≤ m ≲ δ^{−2} be an integer. Let E be a maximal δ-separated subset of the sphere S^{2}, thus #E ∼ δ^{−2}. For let T be a collection of δ × 1 tubes in the ball B(0, C) in R^{3} with directions in E such that for each ω ∈ E there are about m tubes in T oriented in the direction ω. We shall give an informal proof of the fact that^{1}

| ⋃_{T ∈T} T | ≳ m^{1/4}δ^{1/2}.  (1)

This is a consequence of Wolff’s x-ray estimate [5]; note that the earlier Kakeya estimate of Wolff [4] would only give a lower bound on δ^{1/2}. The additional gain in m is essential for the recent argument [1] showing that the Minkowski dimension of Besicovitch sets in R^{3} is strictly greater than 5/2. In higher dimensions (1) has recently been extended [2] to higher dimensions R^{n}, with the corresponding lower bound being m^{1/4}δ^{(n−2)/2}.

As in [4], the idea is to isolate a large “hairbrush” in ⋃_{T ∈T} T, by which we mean a collection of tubes T which all intersect a common stem, and to obtain (1) by obtaining a lower bound on the size of this hairbrush. In [4] the stem of the hairbrush is itself a tube in T, however to obtain (1) we need to use a thicker stem - or more precisely, something called a “plate”.

Let p denote the quantity

p = sup_{δ≤w≤1} sup_{R} \frac{#\{T ∈ T : T ⊂ R\}}{w/δ}  (2)

where R ranges over all Cδ × Cw × C boxes. Note that 1 ≲ p ≲ m.

The quantity p controls the two-dimensional multiplicity of the tubes T in T. More precisely, we have

Lemma 1.1. Suppose N tubes T_{1}, …, T_{N} from T are contained in a Cδ × 1 × 1 slab. Then the total union of these tubes has volume at least Nδ^{2}/p.

Proof This is Cordoba’s argument. Since

∥ ∑_{i=1}^{N} χ_{T_{i}}∥_{1} ∼ Nδ^{2}

1991 Mathematics Subject Classification. 42B15, 35L05.

1Throughout, we shall ignore logarithmic powers of δ.

it suffices by Hölder to show that

∥ ∑_{i=1}^{N} χ_{T_{i}}∥_{2}^{2} ≲ Nδ^{2}p

We write the left-hand side as

∑_{i=1}^{N} ∑_{j=1}^{N} |T_{i} ∩ T_{j}|;

it thus suffices to show that

∑_{j=1}^{N} |T_{i} ∩ T_{j}| ≲ pδ^{2}

for all i.

For any w, consider the contribution of those tubes T_{j} which make an angle of ∼ w with T_{i}. The number of such tubes is at most O(pw/δ) by (2); on the other hand, we have |T_{i} ∩T_{j}| ≲ δ^{3}/w for these times. Putting the two estimates together, and noting that there are only logarithmically many values of δ to worry about, we get the result.

Lemma 1.1 becomes less effective when p is large. To compensate for this we shall make our stem grow with p. More precisely, we use (2) to find a Cδ × Cw × C rectangle R for some δ ≤ w ≤ 1 such that

#{T ∈ T : T ⊂ R} ∼ pw/δ.

Since R can only contain O(w^{2}/δ^{2}) tubes, we see that w ≳ pδ. It turns out that the worst case is when

w ∼ pδ;  (3)

when w is larger, the stem becomes larger and the final estimate for (1) becomes better. So we shall assume (3).

R is now a Cδ × Cpδ × C rectangle containing about p^{2} tubes. By Lemma 1.1 the union of tubes has volume at least pδ^{2}, which is also the volume of R. Hence R is essentially completely filled with tubes.

Define the quantity μ by

| ⋃_{T ∈T} T | = m/μ.  (4)

Another way of saying this is that μ is the average value of the function ∑_{T ∈T} χ_{T}. Heuristically, this means that most points in ⋃_{T ∈T} are contained in about μ tubes; we shall take this heuristic as exact.

The set R can be thought of as the union of about pδ^{−1} δ-balls. Each such ball has about μ tubes from T going through it. If the tubes in T intersect R at a large angle (the small angle case turns out to ultimately give better estimates), then each ball has a completely different set of tubes going through it. Thus R has about μpδ^{−1} tubes from T going through it.

We can foliate space into p-slabs centered around R. If the tubes that intersect R do so at a large angle, then each such tube belongs to exactly one of these slabs. Now suppose we can show the following variant to Lemma 1.1:

Lemma 1.2. Suppose N tubes T_{1}, …, T_{N} from T are contained in a Cpδ × 1 × 1 slab. Then the total union of these tubes has volume at least Nδ^{2}m^{−1/2}p^{−1}.

By splitting up the μpδ^{−1} tubes intersecting R into slabs and applying this lemma to each slab separately, then summing up and pretending that these slabs are disjoint, we obtain

| ⋃_{T ∈T} T | ≳ μpδ^{−1}δ^{2}m^{−1/2}p^{−1}.

Taking the geometric mean of this with (5) we obtain (1).

It remains to show Lemma 1.2. We will prove this assuming that the generic angle of intersection between the tubes T_{i} is about 1; the other cases can be handled by rescaled versions of the following arguments, as in [3].

We now use the original hairbrush argument from [4]. Define the quantity μ′ by

| ⋃_{i=1}^{N} T_{i}| = Nδ^{2}/μ′;  (5)

as before, we pretend that every point in ⋃_{i=1}^{N} T_{i} is contained in about μ′ tubes T_{i}.

A generic tube T_{1} can be thought of as a union of δ^{−1} δ-balls; by the same arguments as before we see that there are about δ^{−1}μ′ tubes T_{i} that intersect T_{1}.

Foliate space into δ-slabs centered at T_{1}. Each of the tubes T_{i} that intersect T_{1} belong to exactly one of these slabs. If one slab contains, say, M tubes, then by Lemma 1.1 this slab contributes at least Mδ^{2}/p to | ⋃_{T ∈T} T |. Adding up all the slabs (which are mostly disjoint) we obtain

| ⋃_{i=1}^{N} T_{i}| ≳ δ^{−1}μ′δ^{2}/p.

Taking the geometric mean of this with (5) we obtain

| ⋃_{i=1}^{N} T_{i}| ≳ N^{1/2}δ^{3/2}p^{−1/2}.

The claim then follows by noting that since the tubes T_{i} are constrained to a pδ-slab and we can only have about m tubes in each direction, that

N ≲ mpδ^{−1}.

References

[1] N. Katz, I. Łaba, T. Tao: An improved bound on the Minkowski dimension of Besicovitch sets in R^{3}, preprint, 1999.

[2] I. Łaba, T. Tao: An x-ray estimate in R^{n}, preprint, 1999.

[3] T. Tao, A. Vargas, L. Vega: A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), 967–1000.

[4] T. Wolff: An improved bound for Kakeya type maximal functions, Revista Mat. Iberoamericana 11(1995), 651–674.

[5] T. Wolff: A mixed norm estimate for the x-ray transform, Revista Mat. Iberoamericana 14(1998), 561–601.

Department of Mathematics, UCLA, Los Angeles, CA 90024

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