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Let $E \subset {\mathbb{R}}^2$ be a compact set with positive $\alpha$-dimensional Hausdorff measure. Without loss of generality, we can suppose that $E$ is contained in the unit disk. Let $E_1$ and $E_2$ be subsets of $E$ with positive $\alpha$-dimensional Hausdorff measure so that the distance from $E_1$ to $E_2$ is $\gtrsim 1$. Each subset $E_i$ admits a measure $\mu_i$ with the following two properties:

$$\label{muprob} \mu_i \textrm{ is a probability measure supported on } E_i.$$

$$\label{mualphadim} \mu_i (B(x, r)) \lesssim r^\alpha.$$

We will explain how to define ${\mu_{1, good}}$ by removing “train-track like” pieces from $\mu_1$. Before going into the details, let us explain the features of a train track that motivate our definition of ${\mu_{1, good}}$. Let $\mu$ be the example in Figure 1 and let $T$ be one of the $R^{-1/2} \times 1$ rectangles containing a train track of the set $E$. One feature of $T$ is that $\mu(T)$ is large. Because the slats of the train track are perpendicular to the direction of $T$, the Fourier transform of $\mu|_T$ is concentrated on frequencies that are in the same direction as $T$. This is a second feature of $T$. So to build ${\mu_{1, good}}$, we will first 