to be the sum of contributions from all the good tubes.

$${\mu_{1, good}}:= M_0 \mu_1 + \sum_{T \in {\mathbb{T}}, T \textrm{ good}} M_T \mu_1.$$

We describe a couple of examples to give a sense of how ${\mu_{1, good}}$ behaves. If $\mu_1$ is the normalized area measure on the set $E$ in Figure 1 above, then ${\mu_{1, good}}$ would be (approximately) the normalized area measure on the union of the large rectangles. On the other hand, if we took the set $E$ in Figure 1 above and we changed it by tilting the slats at a 45 degree angle while keeping the large rectangles vertical, then ${\mu_{1, good}}$ would be essentially equal to $\mu_1$. In general, ${\mu_{1, good}}$ may not be real-valued, but it is a distribution.

Our main theorem (Theorem \[main\]) follows from two estimates about the pushforward measures $d^x_* \mu_1$ and $d^x_* {\mu_{1, good}}$.

\[mainest1\] If $\alpha > 1$, and if we choose $R_0$ large enough, then there is a subset $E_2' \subset E_2$ so that $\mu_2(E_2') \ge 1 - \frac{1}{1000}$ and for each $x \in E_2'$,

$$\| d^x_*(\mu_1) - d^x_*({\mu_{1, good}}) \|_{L^1} < \frac{1}{1000}.$$

\[mainest2\] If $\alpha > 5/4$, then

$$\int_{E_2}  \| d^x_*({\mu_{1, good}}) \|_{L^2}^2 d \mu_2(x) < 