greater than $5/4$. There are two main estimates. The first estimate, in Proposition \[mainest1\], says that for most $x \in E_2$, $\| d^x_*(\mu_{1, bad})\|_{L^1}$ is small, and so the $L^1$ distance between $ d^x_* (\mu_1)$ and $ d^x_*({\mu_{1, good}})$ is small. The bad part, $\mu_{1, bad}$, is made from train track configurations, and that helps us analyze it. Analyzing each individual train track is not difficult. However, unlike in our example above, it could happen that each point lies in many different train tracks going in different directions. To control this type of behavior, we use an estimate of Orponen from [@O17b] which also played a key role in Keleti and Shmerkin’s work on the Falconer problem [@KS18].

The second estimate says that $d^x_* {\mu_{1, good}}$ is better behaved in $L^2$ than $d^x_* \mu_1$. More precisely, Proposition \[mainest2\] says that if $\alpha > 5/4$, then $\int_{E_2} \| d^x_* {\mu_{1, good}}\|_{L^2}^2$ is finite. The proof of Proposition \[mainest2\] is based on Liu’s framework and on decoupling. We will prove and then use a refinement of the decoupling theorem (Theorem \[refdec\]) which is related to the refined Strichartz estimates that appear in [@DGL17], [@DGLZ18], and [@DGOWWZ18]. This refinement of decoupling was proven 