Cardinality of $\eta$-bush; on a Lemma from Wolff's paper.

Question

The Question

This question is about Lemma 1.2 on the fifth page of Thomas Wolff's paper, "A sharp bilinear cone restriction estimate", Annals of Mathematics, 153 (2001), 661--698. The Lemma states (the definitions to be given after)

If $x\in Q(1)$ is a smooth $\mu$-fold point for the white tubes with $\mu > \delta^B$ then $x$ is a base-point for an $\eta$-bush of white tubes with cardinality $\gtrsim (\log \frac1\delta)^{-1} \mu (\frac\eta\delta)^M$ for some $\eta \leq \delta^{1-\epsilon}$. Conversely...

(I understand the converse part, and can sketch the proof, so I omit that here.) The Lemma is given without proof in the paper, and left deliberately as an exercise. The problem is I am not even sure if I understand the Lemma correctly! (In particular, I don't see why the restriction $x\in Q(1)$ is necessary at all: the statement seems to be translation invariant.)

Question Can someone supply a sketch of the proof for this Lemma (beyond the one-liner in the paper)? In particular, where does the logarithm loss come from? The converse statement does not require a logarithm. Thanks.

The Definitions

Now, the definitions to make sense of the Lemma. (Do let me know if I missed anything.)

• $Q(1)\subset \mathbb{R}^d$ is the unit cube centered at the origin.
• A small constant $\epsilon$ is fixed throughout, and a large constant $B$ depending on dimension $d$ is fixed throughout.
• For the purpose of this lemma, a white light-ray is a line in $\mathbb{R}^d$ transversal, and making an angle of 45 degrees, with the plane $\{x_d = 0\}$
• Let $\mathcal{W}$ denote a (finite or countable) collection of white light rays. Fix $\delta > 0$. For an element $W\in \mathcal{W}$, denote by $w$ the set of points $\{ |x - W| \leq \delta\}$. For $w$ the function $\phi_w$ is defined: $ \phi_w(x) = \min (1, \frac{\delta}{|x-w|})^M $ for some fixed large constant $M$ depending on $\epsilon$. $|x-w|$ denotes the Euclidean distance from the point $x$ to the set $w$.
• $\mathcal{W}$ is assumed to be $\delta$ separated: for each $W$ we can associate a direction in real projective space corresponding to the axis. Then for any $D$ disc in projective space of radius $\delta$, let $\mathcal{W}_D$ denote the subset of light rays whose direction points in $D$, the $\delta$-separation assumption requires that the set $\cap_{W\in \mathcal{W}_D} w$ be a bounded set. In particular this implies two parallel light rays cannot be closer than $2\delta$ apart.
• We say that $x$ is a smooth $\mu$-fold point for the white tubes if $\sum_{W\in\mathcal{W}} \phi_w(x) \geq \mu$. Roughly speaking this means around $\mu$ light rays get close to $x$. (As far as I can tell, this "roughly speaking" is the content of the Lemma described above.)
• $x$ is said to be a base-point for an $\eta$-bush $P\subset \mathcal{W}$ if $\sup_P |x-w| \leq \eta$. That is, we have a family of light-rays that all pass within $\delta + \eta$ of $x$.

Answer 1

It looks like a dyadic pigeonholing argument to me (the presence of the logarithm is a big clue in this regard). One can decompose $\phi_w$ into about $\log \frac{1}{\delta}$ dyadic shells, depending on the magnitude of $|x-w|/\delta$, plus a remainder in which $1+|x-w|/\delta \geq \delta^{-100B}$ (say) which has a negligible contribution. So by the pigeonhole principle, $\gtrsim \frac{1}{\log \frac{1}{\delta}} \mu$ of the multiplicity must be coming from one of the shells. Calling the radius of that shell $\eta$, the result presumably follows.

It is quite likely that one could eliminate the logarithm loss here by exploiting the freedom to degrade the M parameter. But in this sort of work, there are logarithmic factors lost all over the place for other reasons, so one more such loss is pretty much insignificant.

Answer 2

Ah, with Terry's comment it turns out the situation was a lot simpler than I thought.

Step 1: Lack of contribution from distant pieces

Divide the region $\{|y-x| > \delta : y \in \mathbb{R}^d\}$ into concentric shells of thickness $\delta$. Consider the contribution to $\sum \phi_w(x)$ from $W\in\mathcal{W}$ such that $|x-w| \in (k\delta, k+1\delta]$, for $k\in \mathbb{N}$. We can divide the shell up into $\sim k^{d-1}\delta^{1-d}$ balls of size $\delta$. Write $y(W)$ for the point of closest approach on $W$ to $x$. In each ball $V$ of size $\delta$, by the $\delta$-separation assumption, there exists sum large constant $B'$ such that

$$ \left|\{ W\in\mathcal{W} : y(W) \in V \}\right| \leq B \delta^{-B'} $$

So

$$ \sum_{k > \delta^{-\epsilon}} \sum_{k\delta < |y(W)-x| - \delta \leq (k+1)\delta} \phi_w(x) \lesssim \sum_{k > \delta^{-\epsilon}} k^{d-1}\delta^{1-d-B'} k^{-M} $$

For sufficiently large $M$ (and if we fix $\delta < \delta_0 < 1$), the RHS can be bounded $\leq \delta^B$. So we can neglect the contribution from far away pieces up to a $\delta^B$ term.

Step 2: Pigeonhole the inside

In view of Step 1, it suffices to show that there exists a constant $C$ such that if for any $0 < k < \delta^{-\epsilon}$ ($k$ should be though of as $\eta/\delta$) we have

$$ \left| \{ W\in\mathcal{W} : |x-w| < k\delta \} \right| \leq C\frac{1}{\log \frac1\delta} \mu k^M $$

we must have $\sum \phi_w(x) < \mu$. This follows by noting that the above means that there are no rays within distance $\delta$ of $x$ ($k = 0$), and thus we only need to sum over $k > 0$

$$ \sum_{k = 1}^{\delta^{-\epsilon}}\phi_w(x) \lesssim \sum_k \left| \{ W\in\mathcal{W} : |x-w| < k\delta \} \right| \cdot k^{-(M+1)} $$

using that $\phi_w(x) \sim k^{-M}$. Plugging in the hypothesis, we have that the expression is controlled by

$$ C\frac{1}{\log \frac1\delta} \mu \sum_{k = 1}^{\delta^{-\epsilon}} k^{-1} \lesssim C \mu \log_\delta(\delta^{-\epsilon}) = C\epsilon \mu$$

So for sufficiently small $C$, we can bound

$$ \sum_{k = 1}^{\delta^{-\epsilon}}\phi_w(x) \leq \frac12 \mu $$

which gives the desired contradiction.