be positive. Liu introduced an interesting way to rewrite this quantity in terms of the Fourier transform of $\mu$. It can then be studied using restriction theory, leading to estimates on the pinned distance problem.

In the planar case, there is an obstruction to pushing either one of these methods to dimensions below $4/3$. For every $\alpha < 4/3$, there is a set $E$ of dimension $\alpha$ and a Frostman measure $\mu$ on $E$ so that $\| d_*(\mu \times \mu) \|_{L^2}$ is infinite, and also $\| d^x_* (\mu) \|_{L^2}$ is infinite for every $x \in E$. This set is a variation on an example from [@KT01]. The set $E$ looks roughly like several parallel train tracks. In the following figure, we show an approximation of the set $E$ at a small scale $R^{-1}$. The measure $\mu$ (approximated at scale $R^{-1}$) is just the normalized area measure on this set.

![image](sketch.pdf) Figure 1

The set $E$ is divided among several large $R^{-1/2} \times 1$ rectangles. Within each of these large rectangles, the set $E$ consists of evenly spaced parallel rectangles with dimensions $R^{-1/2} \times R^{-1}$. Each of these smaller rectangles is called a slat. The restriction of $E$ to one of 