## 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 $x$.