the larger rectangles is called a train track. The spacing between two consecutive slats is controlled by the dimension of $E$, and it works out to $R^{-\alpha/2}$. If $x$ and $y$ are in the same train track, on roughly opposite sides, and if $y$ is $M$ slats from $x$, then $|x-y|$ lies in the interval

$$I_M := [ M R^{-\alpha/2} - R^{-1}, M R^{-\alpha/2} + R^{-1}].$$

Therefore, $ d_*(\mu \times \mu)$ assigns a lot of mass to the union of the intervals $I_M$. This union is quite small, and even though the mass involved is significantly less than 1, it is still enough to force $\int |d_*(\mu \times \mu)|^2$ to be very large.

There is a similar issue for $d^x_*(\mu)$. If we fix any $x \in E$, and we let $T_0$ be the large rectangle containing $x$, then $d^x_* ( \mu|_{T_0})$ is mostly concentrated on $\cup I_M$, and this forces $\int |d^x_* \mu|^2$ to be very large. On the other hand, if $T$ is a large rectangle which is far from $x$, then $d^x_* (\mu|_T)$ is rather evenly distributed – in fact $d^x_*( \mu |_{T})$ is close to the pushforward of the uniform measure on $T$ with the same total 