mass. So if we graph $d^x_* (\mu)$, it has some peaks along $\cup I_M$ coming from the rectangle $T_0$ through $x$, but the bulk of $d^x_*(\mu)$ is spread rather evenly and comes from rectangles $T$ far from $x$. In particular, the support of $d^x_*(\mu)$ indeed has positive Lebesgue measure.

This example is the main obstacle to proving the Falconer conjecture for dimensions less than 4/3. Starting with a general Frostman measure, we separate out a part of it that resembles the train tracks in the example above. Then we estimate the train-track part and the non-train-track part in different ways.

For technical reasons, we consider two subsets $E_1, E_2 \subset E$ separated by distance $\sim 1$, and we let $\mu_1$ and $\mu_2$ be Frostman measures on $E_1, E_2$. In the example above, we can imagine that $E_1$ is the bottom third and $E_2$ is the top third. We divide $\mu_1$ into two pieces

$$\mu_1 = {\mu_{1, good}}+ \mu_{1, bad},$$

where $\mu_{1,bad}$ is essentially the train-track-like part of $\mu_1$. We always arrange, however, that $\int \mu_{1, bad} = 0$.

For example, if $\mu_1$ is the normalized area measure on the set $E_1$ in Figure 1 above, then ${\mu_{1, good}}$ would be 